euclideangeometry (6feb20)

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+%%
+%% This is file `README.txt',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% euclideangeometry.dtx  (with options: `readme')
+%% 
+%%   Copyright (C)  2020 Claudio Beccari all rights reserved.
+%%   License information appended
+%% 
+File README.txt for package euclideangeometry
+        [2020-02-02 v.0.1.2 Extension package for curve2e]
+
+The package bundle euclideangeometry is composed of the following files
+
+euclideangeometry.dtx
+euclideangeometry.pdf
+euclideangeometry-man.tex
+euclideangeometry-man.pdf
+README.txt
+
+euclideangeometry.dtx is the documented TeX source file of package
+euclideangeometry.sty; you get euclideangeometry.sty,
+euclideangeometry.pdf by running pdflatex on euclideangeometry.dtx.
+
+README.txt, this file, contains general information.
+
+euclideangeometry-man.tex and euclideangeometry-man.pdf are
+the source file and the readable document containing the end
+user manual.
+In other words euclideangeometry.pdf is oriented towards the
+developers and euclideangeometry-man.pdf to the end users.
+
+Claudio Beccari
+
+claudio dot beccari at gmail dot com
+%% 
+%% Distributable under the LaTeX Project Public License,
+%% version 1.3c or higher (your choice). The latest version of
+%% this license is at: http://www.latex-project.org/lppl.txt
+%% 
+%% This work is "maintained"
+%% 
+%% This work consists of files:
+%% 1) euclideangeometry.dtx, and the derived files euclideangeometry.sty
+%%                           and euclideangeometry.pdf,
+%% 2) euclideangeometry-man.tex and the derived file euclideangeometry-man.pdf,
+%% 3) the auxiliary derived file README.txt
+%% 
+%%
+%% End of file `README.txt'.
diff --git a/Master/texmf-dist/doc/latex/euclideangeometry/euclideangeometry-man.pdf b/Master/texmf-dist/doc/latex/euclideangeometry/euclideangeometry-man.pdf
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+% !TEX encoding = UTF-8 Unicode
+% !TEX TS-program = pdflatex
+
+\documentclass[11pt,titlepage]{article}\errorcontextlines=100
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage[english]{babel}
+\usepackage{lmodern,textcomp,mflogo}
+\usepackage{amsmath,fancyvrb,graphicx,verbatim,afterpage,trace}
+\usepackage{etoolbox,enumitem,xspace,xcolor,euclideangeometry}
+\usepackage{url}
+
+\AfterEndPreamble{%
+\apptocmd{\TeX}{{}\xspace}{}{}
+\apptocmd{\LaTeX}{{}\xspace}{}{}}
+
+\newcommand\heart[3]{%
+\edef\X{\fpeval{#1*16*(sind(#2)^3)}}
+\edef\Y{\fpeval{#1*(13*cosd(#2) - 5*cosd(2*#2) 
+  - 2*cosd(3*#2) -cosd(4*#2)+2.4)}}
+\CopyVect\X,\Y to#3}
+\let\cuore\heart
+
+\begin{document}
+
+\begin{titlepage}
+\thispagestyle{empty}
+\centering
+\textsc{\large Claudio Beccari}\\[2ex]
+\texttt{claudio dot beccari at gmail dot com}
+
+\vspace{\stretch{2}}
+
+{\LARGE\scshape The \pack{euclideangeometry} package\\
+{\large user manual}\par}
+
+\vspace{\baselineskip}
+
+
+\GetFileInfo{euclideangeometry.sty}
+Version \fileversion\ of \filedate
+
+\vspace*{\stretch{5}}
+\end{titlepage}
+
+ \begin{abstract}
+ This file further extends the functionalities of the \pack{curve2e} 
+ package, which, on turn, is an extension of the \pack{pict2e} package
+ to the standard \env{picture} environment as defined in the \LaTeX\ 
+ kernel source file.
+
+ The \pack{curve2e} package was upgraded a the beginning of 2020; the 
+ material
+ of this new package, might have been included in the former one, but is
+ is sospecific, that we preferred defining a standalone one; this package 
+ takes care of requesting the packages it depends from.
+
+ The purpose is to provide the tools to draw most of the geometrical
+ constructions that a high school instructor or bachelor degree professor
+ might need to teach geometry. The connection to Euclide depends on the
+ fact that in its times calculations were made with ruler, compass, and,
+ apparently, also with ellipsograph,
+
+ The user of this package has available all the machinery provided by
+ the \pack{pict2e} and \pack{curve2e} packages, in order to define new functionalities
+ and build macros that draw the necessary lines, circles, and other such
+ objects, as they would have done in the ancient times. Actually just one 
+ macro is programmed to solve a linear system of equations
+ \end{abstract}
+ 
+ \tableofcontents
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section*{Warning}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+The \pack{euclideangeometry} package requires the advanced functionalities of the \LaTeX\!3 (L3) language; if such functionalities are not available for any reason (incomplete/basic installation of the \TeX system; legacy installation of the \TeX system; the \TeX system has not been updated; \dots) input of this package is stopped, the whole job is aborted, and a visible message is issued. 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section{Introduction}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ The \pack{picture} environment has been available since the very beginning
+ of \LaTeX in 1985. At that time it was a very simple environment
+ that allowed to draw very simple line graphics with many limitations.
+ When \LaTeX was upgraded from \LaTeX\!2.09 to \LaTeXe in 1994, Leslie
+ Lamport announced an upgrade that eventually became available in 2003
+ with package \pack{pict2e}; in 2006 I wrote the \pack{curve2e} package that added
+ many more functionalities; both packages were upgraded during these
+ years; and now line graphics with the \env{picture} environment can perform
+ pretty well. The package \pack{euclideangeometry} adds even more specific
+ functionalities in order to produce geometric drawings as they were
+ possible in the old times, when calculus and analytic geometry were
+ not available.
+ 
+ In these years other drawing programs were made available to the \TeX 
+ community; \pack{PSTricks} and \pack{TikZ} are the most known ones, but there are 
+ other less known packages, that perform very well; among the latter 
+ I would like to mention \pack{xpicture}, that relies on \pack{pict2e} and 
+ \pack{curve2e}, but extends the functionalities with a very smart handling 
+ of coordinate systems, that allow to draw many line drawings suitable 
+ for teaching geometry in high schools and introductory courses in the 
+ university bachelor degree programs.
+ 
+ This package \pack{euclideangeomery} in a certain way follows the same
+ path of \pack{xpicture} but it avoids defining a new user language
+ interface; rather it builds new macros by using the same philosophy of
+ the recent \pack{curve2e} package.
+ 
+ It is worth mentioning that now \pack{curve2e} accepts coordinates in both 
+ cartesian and  polar form; it allows to identify specific points of the 
+ drawing with macros, so the same macro can be used over and over again to 
+ address the same points.The package can draw lines, vectors, arcs 
+ with no arrow tips, or with one arrow tip, or with arrow tips at both ends,
+ arcs included. The macros for drawing poly lines, polygons, circles,
+ generic curves (by means of Bézier cubic or quadratic splines) are
+ already available; such facilities are well documented and exemplified
+ in the user manual of \pack{curve2e} package. 
+
+ In what follows there will be several figures drawn with this package;
+ in the background there is a red grid where the meshes are
+ 10\,\cs{unitlenth} apart in both directions; they should help to
+ understand the position of the various drawings on the picture canvas.
+ This grid is useful also to the end user, while s/he is working on a 
+ particular drawing, but when the drawing is finished, the user can 
+ delete the grid command or comment ot that line of code.
+ For what regards the commands used to render the images, their codes can
+ be found in the documented code file \pack{euclideangeometry.pdf}.
+ 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section{Installing \pack{euclideangeometry}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ You are not supposed to manually install package \pack{euclideangeometry}.
+ In facts you have to work with a complete and updated/upgraded \TeX 
+ installation, otherwise this package won't work; this means that
+ you have done your updating after 2020-01-18. And this package is already
+ present in any modern updated complete installation of the \TeX system.
+ Nevertheless the package will load \pack{curve2e} with the wrong version
+ and file date, but this package will abort its own loading.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section{Loading \pack{euclideangeometry}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ If you want to use the \pack{euclideangeometry} package, we suggest you load it with the following command:
+\begin{flushleft}\obeylines
+\cs{usepackage}\oarg{options}\Marg{euclideangeomery}
+\end{flushleft}
+ The package will take care of managing the possible \meta{options}
+ and to call \pack{curve2e} with the specified options; on turn
+ \pack{curve2e} calls \pack{pict2e} passing on the \meta{options}; such
+ \meta{options} are only those usable by \pack{pict2e} because neither
+ \pack{curve2e} nor \pack{euclideangeometry} use any option. If the
+ user is invoking \pack{euclideangeometry}, it is certain s/he does
+ not want to use the native picture environment, but the modern
+ extended one; therefore the only meaningful possible options are
+ \opt{latex} and\opt{pstricks}; such options influence only the shape
+ of the arrow tips; with option \opt{latex} they are triangular,
+ while with \opt{pstricks} they have the shape of a stealth
+ aircraft. The difference is very small; therefore we imagine that
+ even if these options are available, they might never be used.
+
+ Nothing happens if the user forgets this mechanism; therefore if s/he
+ loads \pack{curve2e} and/or \pack{pict2e}, before
+ \pack{euclideangeomentry} the only problem that might arise is an
+ “Option clash” error message; if two of these packages are selected
+ with different arrow tips; not impossible, of course, by we deem
+ it very unlikely.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section{Available commands}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ The commands available with the first extension \pack{pict2e} to the native
+ \amb{picture} environment, maintain their names but do not maintain the
+ same restrictions; in particular there are the following improvements.
+\begin{enumerate}[noitemsep]
+ \item Lines and vectors are drawn as usual by \cs{put}ting in place
+ their forms, but their inclinations are not limited to a limited
+ number of slope parameters, originally specified with reciprocally
+ prime single digit values not exceeding~6 for lines, and~4 for vectors;
+ the length of these sloped objects is still their horizontal component;
+ now, the slopes may be described with any signed fractional number not
+ exceeding $2^{30}-1$ in absolute value; it still is a limited number of
+ slopes, but their combinations are practically countless.
+%
+\item There is no restriction on the minimum length of lines and vectors.
+%
+\item Circles and dots can be drawn at any size, not at that dozen
+ or so of finite sizes that were accepted with the original environment.
+%
+\item Ovals may be specified the corner curvature; the default
+ size of the quarter circles that make up the oval corners may be
+ specified; if no specification is given the radius of such corners
+ is the maximum that can be fitted in the oval; in practice it is
+ half the shortest value between the oval height and width.
+
+\item The quadratic Bézier splines do not require the specification
+ of the number of dots that were used by the native environment to
+ draw “arbitrary” curves; now they are drawn with continuous curved
+ lines.
+\end{enumerate}
+ Some new commands were added by \pack{pict2e}
+\begin{enumerate}[noitemsep]
+
+\item The third degree (cubic) Bézier splines are sort of new; certainly
+ now they are traced with continuous lines; if it is desired, it is
+ possible to replace the continuous line with a number of dots so as
+ to have a (unevenly) dotted curve. It suffices to specify the number 
+ of dots the curve should be made with.
+
+\item\cs{arc} and \cs{arc*} draw an arc or a filled circular sector,
+ with their centers at the axes origin; therefore  they need to be put
+ in place somewhere else by means of the usual \cs{put} command.
+
+\item The new command \cs{Line} traces a segment from one given point
+ to another point; it is very convenient to specify the end points
+ instead of the slope the line must have to go form the starting
+ point to the ending one. The command does not require the \cs{put}
+ command to put the segment in place; nevertheless it can be shifted
+ somewhere else with \cs{put} if it becomes necessary.
+
+\item the new command \cs{polyline} draws a sequence of connected
+ segments that form a piecewise linear “curve”; the way segments are
+ joined to one another depend from the “join” specifiers that
+ \pack{pict2e} has introduced; they will be described further on.
+
+\item \cs{polygon} and \cs{polygon*} produce closed paths as it would
+ be possible when using \cs{polyline} and specifying the last point
+ coincident with the first point of that curve. The closed path is
+ filled with the default color if the asterisk is used.
+\end{enumerate}
+
+ There were also the low level commands user interfaces to the various 
+ drivers; these drivers really exist, but \pack{pict2e}
+ knows how to detect the correct language of the necessary drive;
+ the user is therefore allowed to pretend to ignore the existence
+ of such drivers; s/he can simply use these commands; their names are
+ almost self explanatory.
+\begin{enumerate}[noitemsep]
+\item \cs{moveto} Sets the start of a line tracing to an initial point.
+
+\item \cs{lineto} traces a segment up to a specified point.
+
+\item \cs{curveto} traces a third degree Bézier up to the third specified
+ point, while using the other two ones as control points.\footnote{If
+ these terms are unfamiliar, please read the \pack{pict2e} documentation.}
+
+ \item \cs{circlearc} traces a circumference arc from the last line
+ point to a specified destination; its center, its angle amplitude, its
+ initial point are among the specified arguments, but the reader should
+ check on the \pack{pict2e} documentation for the details.
+
+\item[]\mbox{\bfseries Attention!} Notice that these commands produce
+ just information to trace lines, but by themselves they do not trace 
+ anything; in order to actually trace the curve or do other operations 
+ with what has been done after the user finished describing the line to 
+ trace, the following low level commands must be used.
+
+\item A \cs{closepath} is necessary if it is desired to
+ join the last position to the initial one. But if the last point
+ specified coincides with the very first one, a closed loop is
+ effectively already completed.
+
+\item If a \cs{strokepath} command is used the line is drawn.
+
+\item If a \cs{fillpath} command is used, the line loop is filled by
+ the current color. Notice, if the described line is not a closed loop,
+ this filling command acts as if the line first point and last point were
+ joined by a straight line.
+
+\end{enumerate}
+
+ While describing a line with the above low level commands, or with
+ the previous high level commands, lines and segments join and finish
+ as described hereafter; the following commands must be used, possibly
+ within a group, before actually tracing a specific line made up with
+ several joined lines or curves. Notice that their effect is just
+ visible with lines as thin as 1\,pt, and very visible with thicker
+ lines.
+\begin{enumerate}[noitemsep]
+\item \cs{buttcap} truncates each line with a sharp cut perpendicular
+ to the line axis exactly through the line end point (default).
+
+\item \cs{roundcap} adds a semicircle to the very end of each line.
+
+\item \cs{squarecap} adds the half square to the very end of each line.
+
+\item \cs{miterjoin} joins two (generally straight) lines with a miter
+ (or mitre) joint; this means that the borders of the line are prolonged
+ until they meet; it is very nice when the junction angle is not far away
+ from, or is larger than $90^\circ$. Apparently for \pack{pict2e} this
+ type of joint is the default.
+
+\item \cs{roundjoin} joins each (generally straight) line with
+ a \cs{roundcap}; it is good in most circumstances.
+
+\item \cs{beveljoin} joins two (generally straight) lines with a miter
+ joint truncated with a sharp cut perpendicular to the bisector of the
+ lines axes; with acute angles it is better than the miter joint, but
+ when angles are very small, even this joint is not adequate.
+\end{enumerate}
+
+ Notice that \cs{buttcap} is the default, but in general it might be
+ better to declare the \cs{roundcap} for the whole document.
+
+ We do not go further in the description of the new \pack{pict2e}
+ modified and new new commands; the reader unfamiliar with programmable
+ drawing and the \pack{pic2e} extensions can consult that package
+ documentation. Actually all commands have been redefined or modified
+ by \pack{curve2e} in order to render them at least compatible with
+ both the cartesian and polar coordinates. In oder to have a better understanding of these details, see figure~\ref{fig:joins}\footnote{The \cs{polyline} macro has the default join of type bevel; remember to specify a different join type if you want a different one.}.
+ 
+ \begin{figure}[!htb] \centering
+ \makebox[\textwidth]{\unitlength=0.009\textwidth
+ \begin{picture}(130,50)
+ \GraphGrid(130,50)
+ \linethickness{4mm}
+ \buttcap\miterjoin
+ \put(10,10){\polyline[\miterjoin](0,0)(30,0)(30,30)
+ \put(0,-10){\makebox(30,10)[b]{\cs{buttcap},\cs{miterjoin}}}}
+ \roundcap\roundjoin
+ \put(50,10){\polyline[\roundjoin](0,0)(30,0)(30,30)
+ \put(0,-10){\makebox(30,10)[b]{\cs{roundcap},\cs{roundjoin}}}}
+ \squarecap\beveljoin
+ \put(90,10){\polyline(0,0)(30,0)(30,30)
+ \put(0,-10){\makebox(30,10)[b]{\cs{squarecap},\cs{beveljoin}}}}
+ \end{picture}}
+ \caption{Different caps and joins}\label{fig:joins}
+ \end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%>>>
+\section{\pack{curve2e} extensions}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ Again we do not enter into the details, because the user can read the
+ new user manual \file{curve2e-manual.pdf} simply by entering and
+ executing the \verb|texdoc curve2e-manual| command into a terminal or 
+ command prompt window; this new manual is available with version~2.2.0 
+ (or higher) of \pack{curve2e} and it contains the extensions and sample 
+ codes for (simple) sample drawings; some examples are not so simple, but 
+ show the power of this package upgrade.
+
+ The most important two changes are $(a)$ the choice of different
+ coordinates for addressing points on the drawing canvas, and $(b)$
+ the possibility of using macros to identify specific points.
+ As already mentioned, such changes have been applied also to most,
+ if not all\footnote{I assume I have upgraded all such commands; if not,
+ please, send me a bug notice; I will acknowledge your contribution.}
+ commands defined by \pack{pict2e}.
+ 
+ \pack{curve2e} defines a lot of operations the user can do with
+ the point coordinates; this is done by assuming they are complex
+ numbers, or vectors, or roto\-amplification operators, and making
+ with such entities a lot of actions compatible with their “incarnation".
+ For example multiplying a vector by a roto\`amplification operator,
+ in spite the fact that internally they are both represented by ordered
+ pairs of (generally) fractional numbers, means simply obtaining a new
+ vector rotated and scaled with respect to the original one; the point
+ addressed by the first vector, becomes another point in a different
+ precise position.
+
+ Below you see several examples of usage of such commands; but here space
+ will be saved if a short list is made concerning these “complex number”
+ operations.
+ 
+ Remember the double nature of such complex numbers:
+\[
+z = x + \iu y = m\eu^{\iu \phi}
+\]
+ therefore addition and subtraction are simply done with
+\[
+z_1 \pm z_2 = x_1 \pm x_2 +\iu (y_1 \pm y_2)
+\]
+ Multiplications and divisions are simply done with
+\begin{align*}
+z_1 z_2 &=( m_1 m_2) \eu^{\iu(\phi_1 + \phi_2)} \\
+z_1/z_2 &=( m_1/m_2) \eu^{\iu(\phi_1 - \phi_2)}
+\end{align*}
+ Squares and square roots\footnote{The square root of a complex number 
+ has two values; here we do not go into the details on how \pack{curve2e} 
+ choses one or the other value. In practice, the \pack{curve2e} macros 
+ that use square roots, work mostly on scalars to find magnitudes that 
+ are always positive.} are simply done with:
+\begin{align*}
+z^2      &= m^2\eu^{\iu 2\phi}\\
+\sqrt{z} &= \sqrt{m}\eu^{\iu\phi/2}
+\end{align*}
+ The complex conjugate of a complex number is shown with a superscript
+ asterisk:
+\[\text{if } z= x+\iu y \text{ then } z^\star = x- \iu y\]
+ and from these simple formal rules many results can be obtained;
+ and therefore several macros must be defined.
+
+ But let us summarise. Here is a short list with a minimum of
+ explanation of the commands functionalities introduced by
+ \pack{curve2e}. The user notices that many commands rely on
+ a delimited argument command syntax; the first arguments can
+ generally be introduced with point macros, as well as numerical
+ coordinates (no matter if cartesian and polar ones) while
+ the output(s) should always be in form of point macro(s). Parentheses
+ for delimiting the ordered couples or the point macros are seldom
+ required. On the other side, the variety of multiple optional
+ arguments, sometimes requires the use of different delimiters,
+ most often than not the signs~\texttt{<~>}, in addition to the
+ usual brackets. These syntax functionalities are available with the
+ \pack{xparse} and \pack{xfp} packages, that render the language~L3
+ very useful and effective.
+
+ Handling of complex numbers is done with the following commands.
+ New commands to draw special objects, are also described.
+\begin{enumerate}[noitemsep]
+%
+\item Cartesian and polar coordinates; they are distinguished by
+ their separator; cartesian coordinates are the usual comma separated
+ couple \meta{$x,y$}; polar coordinates are specified with a colon
+ separated couple \meta{$\theta{:}\,\rho$}. In general they are
+ specified within parentheses, but some commands require them without
+ any parentheses. In what follows a generic math symbol, such as for
+ example $P_1$, is used to indicate a complex number that addresses
+ a particular point, irrespective of the chosen coordinate type,
+ or a macro  defined to contain those coordinates.
+
+\item The complex number/vector operations already available with
+ \pack{curve2e} are the following; we specify “macro” because in general
+ macros are used, instead of explicit numerical values, but for input
+ vector macros it is possible to use the comma or colon separated ordered
+ couple; “versor” means “unit vector”; angles are always expressed in
+ degrees; output quantities are everything follows the key word
+ \texttt{to}; output quantities are alway supposed to be in the form
+ of control sequences.
+\begin{itemize}\small
+\item \cs{MakeVectorFrom}\meta{number,number}\meta{numeric macro} to\meta{vector macro}
+\item \cs{CopyVect}\meta{vector macro} to\meta{vector macro}
+\item \cs{ModOfVect}\meta{vector macro} to\meta{modulus macro}
+\item \cs{DirOfVect}\meta{vector macro} to\meta{versor macro}
+\item \cs{ModAndDirOfVect}\meta{vector macro} to\meta{modulus macro}
+    and\meta{versor macro}
+\item \cs{ModAndAngleOfVect}\meta{vector macro} to \meta{modulus macro}
+ and\meta{angle macro}
+\item \cs{DistanceAndDirOfVect}\meta{1st vector macro}
+    minus\meta{2nd vector macro} to\meta{distance macro}
+    and\meta{versor macro}
+\item \cs{XpartOfVect}\meta{vector macro} to\meta{numerical macro}
+\item \cs{YpartOfVect}\meta{vector macro} to\meta{numerical macro}
+\item \cs{DirFromAngle}\meta{angle macro} to\meta{versor macro}
+\item \cs{ArgOfVect}\meta{vector macro} to\meta{angle macro}
+\item \cs{ScaleVect}\meta{vector macro} by\meta{scale factor}
+    to\meta{vector macro}
+\item \cs{ConjVect}\meta{vector macro} to\meta{conjugate vector macro}
+\item \cs{SubVect}\meta{subtrahend vector} from\meta{minuend vector}
+    to\meta{vector macro}
+\item \cs{AddVect}\meta{1st vector} and\meta{2nd vector}
+    to\meta{vector macro}
+\item \cs{Multvect}\marg{1st vector}\meta{$\star$}\marg{2nd vector}\meta{$
+    \star$}\meta{output vector macro}\newline the asterisks are optional;
+    either one changes the \meta{2nd vector} into its complex conjugate
+\item \cs{MultVect}\meta{1st vector}\meta{$\star$}\meta{2nd vector}
+    to\meta{vector macro}\newline discouraged; maintained for backward
+    compatibility; the only optional asterisk changes the \meta{2nd vector}
+    into its complex conjugate
+\item \cs{Divvect}\marg{dividend vector}\marg{divisor vector}\marg{output
+    vector macro}
+\item \cs{DivVect}\meta{dividend vector}\meta{divisor vector}
+    to\meta{vector macro}\newline maintained for backwards
+    compatibility
+\end{itemize}
+
+\item A new command \cs{segment}\parg{$P_1$}\parg{$P_2$} draws a line that
+ joins the specified points.
+
+\item Command \cs{Dashline}\parg{$P_1$}\parg{$P_2$}\marg{dash length}
+ draws a dashed line between the specified points; the
+ \meta{dash length} is specified as a coefficient of
+ \cs{unitlenth} so they are proportioned  to the diagram scale. The gap
+ between dashes is just as wide as the dashes; they are recomputed by
+ the command in order to slightly adjust the \meta{dash length} so
+ that the line starts at point $P_1$ with a dash, and ends at $P_2$
+ again with a dash.
+
+\item Command \cs{Dotline}\parg{$P_1$}\parg{$P_2$}\marg{gap}\oarg{diameter}
+ traces a dotted line between the specified points with dots \meta{gap}
+ units apart, starting and ending with a dot at the specified points.
+ Optionally the absolute diameter of the dots may be specified: a
+ diameter of 1\,pt (default) is visible, but it might be too small;
+ a diameter of 1\,mm is really very black, and may be too large; if
+ the diameter is specified without dimensions they are assumed by
+ default to be typographic points.
+
+\item Command \cs{polyline}, \cs{polygon} and \cs{polygon*} are redefined
+ to accept both coordinate kinds.
+
+\item Commands \cs{VECTOR}\parg{$P_1$}\parg{$P_2$} (and \cs{VVECTOR}, with the same syntax) draw vectors with one arrow tip at the end, or arrow tips at both ends respectively.
+
+\item New commands \cs{Arc}\parg{center}\parg{start}\marg{angle} and,
+ with the same syntax, \cs{VectorArc} and \cs{VectorARC} draw 
+ arcs without or with arrow tip(s), with the specified \meta{center},
+ starting at point \meta{start}, with an aperture of \meta{angle}
+ degrees (not radians). \cs{Arc} draws the arc without arrow tips;
+ \cs{VectorArc} draws the arc with one arrow tip at the end point;
+ \cs{VectorARC} draws an arc with arrow tips at both ends.
+
+\item Command \cs{multiput} has been redefined to accept optional
+ arguments, besides the use of coordinates of both kinds. The new syntax
+ is the following:
+\begin{flushleft}\obeylines
+\cs{multiput}\oarg{shift}\parg{origin}\parg{step}\marg{number}\marg{object}\oarg{handler}
+\end{flushleft}
+ where, if you neglect the first and the last (optional) arguments,
+ you have the original syntax; the \meta{origin} point is where the first
+ \meta{object} is placed; \meta{step} is the displacement of a new
+ \meta{object} relative to the previous one; \meta{number} is the
+ total number of \meta{object}s put in place by the command; possibly
+ the number may be an integer expression computed with the \cs{inteval}
+ function of the L3~language, accessed through the \pack{xfp} package
+ already loaded by \pack{curve2e}. The new features are \meta{shift},
+ that is used to displace the whole drawing somewhere else (in case
+ some fine tuning is required), and \meta{handler}; the latter is a
+ powerful means to control both the object to be set in place and its
+ position; further on there will be examples that show that the object
+ can be put not only on straight paths, but also un other curves,
+ including parabolas, circles, and other shapes.
+
+\item Another version of repetitive commands \cs{xmultiput} is very
+ similar to \cs{multiput} but the iterations are controlled in a
+ different way so that it is possible also to draw continuous
+ curves describing analytical functions even with parametric
+ equations. Further on there will be some examples.
+
+\item The preloaded \pack{xfp} package provides two important 
+ functionalities, i.e. two L3 “functions”, \cs{fpeval} and \cs{inteval};
+ the latter executes expressions on integer numbers containing the usual
+ operators \texttt{+, -, *, /}; the division quotient is rounded to the
+ nearest (positive or negative) integer. The former operates with real
+ fractional numbers and, in addition to the usual arithmetical operators as
+ \cs{inteval}, it can use many mathematical functions, from square roots,
+ to exponentials, logarithms, trigonometric and hyperbolic direct and
+ inverse functions\footnote{The implementation of inverse hyperbolic
+ function is on the L3~Team “to do” list.}, plus other ones. Normally
+ fractional numbers are operated on decimal strings, with 16 fractional
+ places, and 14 integer places but the L3~functions accept also scientific
+ notation. The user can specify truncation or rounding to a specified
+ number of digits. Such integer and fractional mathematical operations
+ are already integrated in most computations performed by \pack{curve2e}.
+
+\item \pack{curve2e} provides two more L3 functions: \cs{fptest} and
+ \cs{fpdowhile} with the following syntax:
+\begin{flushleft}\obeylines
+\cs{fptest}\marg{test}\marg{true}\marg{false}
+\cs{fpdowhile}\marg{test}\marg{actions}
+\end{flushleft}
+ For both macros the \meta{test} is a logical F3~expression; its operands
+ are logical constants, logical values, logical numeric comparisons; its
+ operators are the typical \texttt{||}, \texttt{\&\&}, and \texttt{!},
+ respectively for OR, AND, and NOT. The logical numerical comparisons
+ are mathematical constants or expressions connected with relation
+ operators, such as \texttt{>, =, <}; such operators may be negated with
+ the NOT operator; therefore, for example, \texttt{!>} means “not greater 
+ than”, therefore “lower or equal to”. 
+
+\item The above tests are very useful to control both \cs{fptest} and
+ \cs{fpdowhile}. The logical \meta{test} result lets \cs{fptest} execute
+ only the \meta{true} or the \meta{false} code. Before using \cs{fpdowhile}
+ the \meta{test} expression must be initialised to be \texttt{true};
+ the \meta{actions} should contain some code to be iteratively executed,
+ but they must contain some assignments, typically a change in an
+ iteration counter, such that eventually the \meta{test} logical
+ expression becomes \texttt{false}. Lacking this assignments, the
+ loop continues to infinity, or better, until a fatal error message
+ is issued that informs that the program working memory is exhausted.
+
+\item Such new commands are already used to code the \cs{multiput} and
+ \cs{xmultiput} commands, but they are available also to the user who can
+ operate in a very advanced way; further on, some examples will show
+ some advanced drawings.
+
+\item General curves can be drawn by \pack{pic2e} command \cs{curve}
+ that is sort of difficult to use, because the user has to specify
+ also the control points of the third order Bézier splines. Some other
+ new commands are available with \pack{curve2e}, that are supposed
+ to be easier to use; they are described in the following items.
+
+\item The new command \cs{Curve} joins a sequence of third order
+ splines by simply specifying the node-direction coordinates; i.e. at the
+ junction of two consecutive splines, in a certain interpolation node the
+ final previous spline tangent has the same direction of the tangent
+ at the second spline first node; if a change of direction is required, an
+ optional new direction can be specified. Therefore this triplet of
+ information has the following syntax:
+\begin{flushleft}
+ \parg{node}\aarg{direction}\oarg{new direction}
+\end{flushleft}
+ Evidently the \meta{new direction} is specified only for the nodes
+ that correspond to a cusp. A variation of the command arguments is
+ available by optionally specifying the “looseness” of the curve:
+\begin{flushleft}
+\parg{node}\aarg{direction\texttt{;}start\texttt{,}end}\oarg{\dots}
+\end{flushleft}
+ where \meta{start} is the spline starting “looseness” and \meta{end}
+ is the spline ending one. These (generally different) values
+ are an index of how far is the control point from the adjacent node.
+ With this functionality the user has a very good control on the curve
+ shape and curvature.
+
+\item A similar command \cs{Qurve} works almost the same way, but it
+ traces a quadratic Bézier spline; this one is specified only with two
+ nodes an a single control point, therefore is less configurable than
+ cubic splines; the same final line requires several quadratic splines
+ when just a single cubic spline might do the same job. Notice also that
+ quadratic splines are just parabolic arcs, therefore without inflections,
+ while a cubic spline can have one inflexion. 
+
+\item A further advanced variation is obtained with the new
+ \cs{CurveBetween} command that creates a single cubic spline between two
+ given points with the following syntax:
+\begin{flushleft}\ttfamily
+\cs{CurveBetween}\meta{node1} And\meta{node2} WithDirs \meta{dir1}
+    And\meta{dir2}
+\end{flushleft}
+
+\item A similar variant command is defined with the following syntax:
+\begin{flushleft}\ttfamily
+ \cs{CbezierBetween}\meta{node1} And\meta{node2} WithDirs\meta{dir1}
+    And\meta{dir2}\\ 
+\qquad UsingDists\meta{dist1} And\meta{dist2}
+\end{flushleft}
+ Usage examples are shown in section~\ref{sec:examples}
+\end{enumerate}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{ Euclidean geometry commands}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ With the already large power of \pack{curve2e} there was a push towards
+ specialised applications; the first of which was, evidently, geometry;
+ that kind of geometry that was used in the ancient times when
+ mathematicians did not have available the sophisticated means they
+ have today; they did not even have a positional numerical notation, that
+ arrived in the “west” of the world we are familiar with, just by
+ the XI-XII century; before replacing the roman numbering system another
+ couple of centuries passed by; real numbers with the notation we use
+ today with a decimal separator, had to wait till the XVI century (at
+ least); many things that naw are taught in elementary school were
+ still a sort of magic until the end of XVIII century.
+
+ Even a simple algebraic second degree equation was a problem.
+ In facts the Renaissance was the artistic period when the classical
+ proportions were brought back to the artists who could not solve the
+ simple equation where a segment of unit length is divided in two
+ unequal parts $x$  and $1-x$ such that the following proportion exists
+ among the various parts and the whole segment:
+\[
+\frac{x}{1} = \frac{1-x}{x} \Longrightarrow x = \frac{1}{x} - 1
+\]
+ today we cam solve the problem by manipulating that simple proportion
+ to get
+\[
+ x^2 + x -1 = 0
+\]
+ and we know that the equation has two solution of opposite signs, and that
+ their magnitudes are the reciprocal of one another. Since we are
+ interested in their magnitudes, we adapt the solutions in the form
+\begin{equation}
+ x_{1,2} = \frac{\sqrt{5} \pm 1}{2}= \sqrt{1+0.5^2}\pm 0.5 \Longrightarrow
+\begin{cases}x_1=1.618\ldots\\x_2=0.618\ldots \end{cases}\label{equ:aureo}
+\end{equation}
+ The larger number is called the \emph{golden number} and the smaller
+ one the \emph{golden section}. 
+
+ Luca Pacioli, by the turn of centuries XV–XVI, was the tutor of
+ Guidubaldo, the heir of Federico di Montefeltro, Duke of
+ Urbino\footnote{If you never visited this Renaissance city and its Ducal
+ Palace, consider visiting it; it is one of the many UNESCO Heritage
+ places.}; he wrote the famous book \emph{De Diuina Proportione} that
+ contained also the theory of the golden section accompanied by beautiful
+ drawings of many Platonic solids and other non convex ones, drawn by
+ Leonardo da Vinci. Everything was executed with perfect etchings, even the
+ construction of the golden section; in its basic form\footnote{The third
+ formula in equation~\eqref{equ:aureo} is written in such a way as to
+ explain the graphical construction in figure~\ref{fig:aureo}.} it is
+ replicated in figure~\ref{fig:aureo}. 
+\begin{figure}
+\begin{Esempio}[\setfontsize{7}](0.40)
+\unitlength=0.005\linewidth
+\begin{picture}(170,140)(0,-70)
+\GraphGrid(0,-70)(170,140)
+\VECTOR(0,0)(170,0)
+\Pbox(170,0)[t]{x}[0]
+\Pbox(100,0)[t]{\mathrm{1}}[2]
+\Pbox(0,0)[r]{O}[2]
+\Arc(100,0)(50,0){-90}
+\segment(100,0)(100,70)
+\segment(0,0)(100,50)
+\Pbox(50,0)[tr]{\mathrm{0.5}}[2]
+\ModAndAngleOfVect100,50 to\M and\A 
+\Arc(0,0)(\M,0){\A}\Pbox(\M,0)[bl]{C}[2]
+\Arc(\M,0)(\M,-50){90}
+\Arc(\M,0)(\M,-50){-90}
+\Pbox(\fpeval{\M-50},0)[b]{\mathit{x_2}}[3]
+\Pbox(\fpeval{\M+50},0)[b]{\mathit{x_1}}[3]
+\put(\M,0){\Vector(-70:50)}
+\Pbox(120,-25)[bl]{\mathit{r}=\mathrm{0.5}}[0]
+\thicklines
+\segment(0,0)(100,0)
+\end{picture}
+\end{Esempio}
+\caption{The golden section $x_2$ and the golden number $x_1$}\label{fig:aureo}
+\end{figure}
+ By the way figure~\ref{fig:aureo} shows also the code that is used
+ for the drawing done completely with the facilities available just with \pack{curv2e}. It is also a usage example of several commands. 
+
+ Illiteracy was very widespread; books were expensive and were
+ common just in the wealthy people mansions.
+
+ Mathematicians in the classical times B.C. up to the artists in the
+ Renaissance, had no other means but to use geometrical constructions with
+ ruler and compass. Even today in schools where calculus is not yet
+ taught as a normal subject, possibly not  in certainly high school degree
+ courses, but certainly not in elementary and junior high schools, the
+ instructors have to recourse to geometrical constructions. Sometimes, as
+ in Italy, access to public universities is open with no restrictions to
+ all students with a high school diploma for degree courses that are 
+ more vocational than cultural. Therefore such students in some university 
+ degree courses have to frequent upgrading courses in order to master some 
+ more mathematics compared to what they studied during their basic 
+ education.
+
+ The instructors nowadays very often prepare some booklets with their
+ lessons; such documents, especially in electronic form, are a nice help
+ for many students. And \LaTeX is used to write such documents.
+ Therefore this extension module is mostly dedicated to such instructors.
+
+ The contents of this module is not exhaustive; it just shows a way to
+ use the \pack{curve2e} facilities to extend it to be suited for the
+ kind of geometry they teach.
+ 
+ Here we describe the new commands provided by this package; then in
+ section~\ref{sec:examples} we show their usage by means  examples.
+
+\begin{enumerate}[noitemsep]
+\item Command \cs{IntersectionOfLines} is a fundamental one; its syntax is
+ the following:
+\begin{ttsintassi}
+\cs{IntersecionOfLines}\parg{point1}\parg{dir1} and\parg{point2}\parg{dir2} to\meta{vector}
+\end{ttsintassi}
+ were each line is identified with its \meta{point} and its direction
+ \meta{dir}; the intersection coordinates go to the output \meta{vector}.
+
+\item A second command\cs{IntersectionOfSegments} does almost the same
+ work, but the coordinates of a segment define also its direction,
+ which is the argument of the difference of the terminal nodes of each
+ segment; the syntax therefore is the following:
+\begin{ttsintassi}
+\cs{IntersectionOfSegments}\parg{point11}\parg{point12} 
+ and\parg{point21}\parg{point22}to\meta{vector}
+\end{ttsintassi}
+ Again the intersection point coordinates go to the output \meta{vector}.
+ The first segment is between points 11 and 12, and, similarly, the second
+ segment is between points 21 and 22.
+
+\item Command \cs{ThreePointCircle} draws a circle that goes through three
+ given points; the syntax is the following:
+\begin{ttsintassi}
+ \cs{ThreePointCircle}\meta{$\star$}\parg{point1}\parg{point2}\parg{point3}
+\end{ttsintassi}
+ A sub product of this macro is formed by the vector \cs{C} that contains
+ the coordinates of the center of the circle, that might be useful even
+ if the circle is not drawn; the optional asterisk, if present, does not
+ draw the circle, but the center is available.
+
+\item Alternatively
+\begin{ttsintassi}
+ \cs{ThreePointCircleCenter}\parg{point1}\parg{point2}\parg{point3}to\meta{vector}
+\end{ttsintassi}
+ computes the three point circle center assigning its coordinates to
+ \meta{vector}.
+
+\item Command \cs{CircleWithCenter} draws a circle given its center and it
+ radius; in facts the syntax is the following:
+\begin{ttsintassi}
+\cs{CircleWithCenter}\meta{center} Radius\meta{Radius}
+\end{ttsintassi}
+ This macro does not require the \cs{put} command to put the circle
+ in place.
+
+\item A similar macro \cs{Circlewithcenter} does almost the same; its
+ syntax is the following:
+\begin{ttsintassi}
+\cs{Circlewithcenter}\meta{center} radius\meta{radius}
+\end{ttsintassi}
+ Apparently these two commands do the same, but, no, they behave
+ differently: in the former command the \meta{Radius} is a vector the
+ modulus of which si computed and used as the radius; in the latter
+ command the \meta{radius} is a scalar and (its magnitude) is directly
+ used.
+
+\item Command with syntax:
+\begin{ttsintassi}
+\cs{AxisOf}\meta{point1} and\meta{point2} to \meta{point3} and\meta{point4}
+\end{ttsintassi}
+ is used to determine the axis of a segment; the given
+ segment is specified with its end points \meta{point1} and \meta{point2}
+ and the axis is determined by point \meta{point3} and \meta{point4};
+ actually \meta{point3} is the middle point of the given segment.
+
+\item These two commands with syntax:
+\begin{ttsintassi}
+\cs{SegmentCenter}\parg{point1}\parg{point2}to\meta{center}
+\cs{MiddlePointOf}\parg{point1}\parg{point2}to\meta{center}
+\end{ttsintassi}
+ determine just the middle point between two given points. They are
+ totally equivalent, aliases to one another; sometimes it is more
+ convenient to use a name, sometimes the other; it helps reading the
+ code and maintaining it.
+
+\item Given a triangle  and a specific vertex, it is possible to determine
+ the middle point of the opposite side; it is not very difficult, but it
+ is very handy to have all the necessary elements to draw the median line.
+ The simple syntax is the following:
+\begin{ttsintassi}
+\cs{TriangleMedianBase}\meta{vertex} on\meta{base1} and\meta{base2}
+\qquad to\meta{base middle point}
+\end{ttsintassi}
+
+\item A similar command \cs{TriangleHeightBase} is used to determine the
+ intersection of the height segment from one vertex to the opposite base;
+ with triangles that have an obtuse angle, the height base might lay
+ externally to one of the bases adjacent to such an angle. The syntax is
+ the following
+\begin{ttsintassi}
+\cs{TriangleHeigthtBase}\meta{vertex} on\meta{base1} and\meta{base2} to\meta{height base}
+\end{ttsintassi}
+
+\item Similarly there is the \cs{TriangleBisectorBase} macro with
+ a similar syntax:
+\begin{ttsintassi}
+\cs{TriangleBisectorBase}\meta{vertex} on\meta{base1} and\meta{base2}
+\qquad to\meta{bisector base}
+\end{ttsintassi}
+
+\item A triangle \emph{barycenter} is the point where its median lines
+ intersect; command \cs{TriangleBarycenter} determines its coordinates
+ with the following syntax.
+\begin{ttsintassi}
+\cs{TriangleBarycenter}\parg{vertex1}\parg{vertex2}\parg{vertex3} to\meta{barycenter}
+\end{ttsintassi}
+
+\item A triangle \emph{orthocenter} is the point where its height lines
+ intersect; command \cs{TriangleOrthocenter} determines its coordinates
+ with the following syntax:
+\begin{ttsintassi}
+\cs{TriangleOrthocenter}\parg{vertex1}\parg{vertex2}\parg{vertex3} to\meta{orthocenter}
+\end{ttsintassi}
+
+\item A triangle \emph{incenter} is the point where its bisector lines
+ intersect; command \cs{TriangleIncenter} determines its coordinates
+ with the following syntax:
+\begin{ttsintassi}
+\cs{TriangleIncenter}\parg{vertex1}\parg{vertex2}\parg{vertex3} to\meta{incenter}
+\end{ttsintassi}
+
+\item The distance of a specified point from a given segment or line is
+ computed with the following command
+\begin{ttsintassi}
+\cs{DistanceOfPoint}\meta{point} from\parg{point1}\parg{point2} to\meta{distance}
+\end{ttsintassi}
+ where \meta{point} specifies the point and \meta{point1} and \meta{point2}
+ identify two points on a segment or a line; \meta{distance} is a scalar
+ value.
+
+\item In a construction that will be examined in section~\ref{sec:examples}
+ we need to determine an ellipse axis if the other axis and the focal
+ distance are know; actually it solves the relation
+\begin{equation}a^2= b^2 +c^2\label{equ:axes-foci}\end{equation}
+ that connects such three quantities; $a$ is always the largest of the
+ three quantities; therefore the macro tests if the first entry is larger
+ than the second one: if is is, it computes a Pitagorean difference,
+ otherwise the user should pay attention to use as the first entry the
+ smaller among $b$ and $c$, so as to compute a Pitagorean sum. The command
+ is the following:
+\begin{ttsintassi}
+\cs{AxisFromAxisAndFocus}\meta{axis or focus} and\meta{focus or axis} 
+\qquad to\meta{other axis or focus}
+\end{ttsintassi}
+ The word “axis” stands for “semi axis length”; the word “focus" stands
+ for “focal semi distance”; actually the macro works equally well with
+ full lengths, instead of half lengths; its is important not to mix
+ full and half lengths. Such lengths are expressed as factors of
+ \cs{unitlength}, not as absolute values. This command is described 
+ again when dealing with the specific problem referred to at the 
+ beginning of this list item;  the description is going to be more 
+ detailed and another macro is added to avoid possible errors.
+
+\item Given a segment, i.e. the coordinates of its end points, it is
+ useful to have a macro that computes its length; at the same time it is
+ useful to to compute its direction; this operation is not the same as to
+ compute modulus and argument of a vector, but consists in computing such
+ quantities from the difference of the vectors pointing to the segment
+ end points. These two macros are the following:
+\begin{ttsintassi}
+\cs{SegmentLength}\parg{point1}\parg{point2} to\meta{length}
+\cs{SegmentArg}\parg{point1}\parg{point2} to\meta{argument}
+\end{ttsintassi}
+ The \meta{argument} is computed in the interval $-180^\circ < \phi \leq
+ +180^\circ$; it represents the argument of the vector that goes from
+ \meta{point1} to \meta{point2}, therefore the user must pay attention to
+ the order s/he enters the end points coordinates.
+
+\item The next command \cs{SymmetricalPointOf} is used to find the
+ reflection of a specified point with respect to a fixed point; of course
+ the latter is the middle point of the couple, but the unknown to be
+ determined is not the center of a segment, but one of its end points.
+ The syntax is the following:
+\begin{ttsintassi}
+\cs{SymmetricalPointOf}\meta{point1} respect\meta{fixed} to\meta{point2}
+\end{ttsintassi}
+
+\item  Command \cs{RegPolygon} draws a regular polygon inscribed within
+ a circle of given radius and center, with a specified number of sides;
+ optional arguments allow to  specify color and thickness of the sides,
+ or the polygon interior color; this macro operates differently from the 
+ one for drawing ellipses, that draws simultaneously an ellipse with
+ the border of a color and the interior of another one; with this macro the
+ user who wants to achieve this effect must superimpose to polygons with
+ different settings; but it would not be too difficult to arrange a new
+ macro or to modify this one in order to get “bicolor” polygons.
+ It is not necessary for the purpose of this package, therefore we
+ let the user express his/her phantasy with other macros. The actual
+ syntax is the following:
+\begin{ttsintassi}
+\cs{RegPolygon}\meta{$\star$}\parg{center}\marg{radius}\marg{sides}\oarg{angle}\aarg{settings}
+\end{ttsintassi}
+ The initial optional asterisk specifies if the interior has to be
+ coloured; if yes, the \meta{settings} refer to the color of the
+ interior; if not, the \meta{settings} refer to the thickness and
+ color of the sides; no \meta{settings} imply sides drawn with the
+ default line thickness, generally the one corresponding to \cs{thinlines},
+ and the default color (generally black) for the sides or the interior.
+ By default the first vertex is set to an angle of $0^\circ$ with
+ respect to the \meta{center}; the optional \meta{angle} modifies
+ this value to what is necessary for a particular polygon. The
+ \meta{center} itself is optional, in the sense that if it is not
+ specified the center lays in the origin of the \env{picture} axes; if this 
+ argument is specified, the polygon center is displaced accordingly.
+ The number of sides in theory may be very high, but it is not wise
+ to exceed a couple of dozen sides; if the number of sides is too
+ high, the polygon becomes undistinguishable from a circumference.
+
+\item Several macros are dedicated to ellipses; their names are spelled
+ in Italian, “ellisse”, because the name “ellipse” is already taken by
+ other packages; with Italian user command names there should be no
+ interference with other packages, or the risk is reduced to
+ a minimum. The various macros are \cs{ellisse}, \cs{Sellisse},
+ \cs{Xellisse}, \cs{XSellisse}, \cs{EllisseConFuoco} \cs{EllisseSteiner};
+ the last two control sequence names are aliased with the corresponding
+ English ones \cs{EllipseWithFocus} and \cs{SteinerEllipse}. For the
+ other four ones it is wise to avoid English names for the reasons
+ explained above. After all the Italian  and the English names are very
+ similar and are pronounced almost identically.
+
+ {\tolerance=3000 Actually \cs{ellisse} is practically a shorthand for 
+ \cs{Sellisse} because
+ some optional arguments are already fixed, but the meaning of
+ \cs{fillstroke} depends on the presence or absence of an initial
+ asterisk; similarly \cs{Xellisse} is a sort of a shorthand for
+ \cs{XSellisse}; in facts those commands, that contain
+ an ‘S’ in their names, can optionally perform also the affine
+ \emph{shear} transformation, while those without the ‘S’ do not execute
+ such transformation. Figure~\ref{fig:shear} displays a normal ellipse
+ with its bounding rectangle, and the same ellipse to which the shear
+ affine transformation is applied; the labeled points represent the
+ third order Bézier spline nodes and control points.\par}
+\begin{figure}[!htb]
+\dimendef\Wmp=2000 \Wmp=\dimexpr(\textwidth-\columnsep)/2\relax
+\begin{minipage}{\Wmp}\centering
+\unitlength=0.007\linewidth
+\begin{picture}(100,70)(-50,-35)
+\XSellisse(0,0){50}{25}*
+\end{picture}
+\end{minipage}
+\hfill
+\begin{minipage}{\Wmp}\centering
+\unitlength=0.007\linewidth
+\begin{picture}(100,70)(-50,-35)
+\XSellisse(0,0)<20>{50}{25}*
+\end{picture}
+\end{minipage}
+\caption{The effect of shearing an ellipse with its bounding rectangle}\label{fig:shear}
+\end{figure}
+
+\item The syntax of those six commands are the following:
+\begin{ttsintassi}
+\cs{Sellisse}\meta{$\star$}\marg{semiaxis-h}\marg{semiaxis-v}\oarg{shear}
+\cs{ellisse}\meta{$\star$}\marg{semiaxis-h}\marg{semiaxis-v}
+\cs{XSellisse}\meta{$\star$}\parg{center}\oarg{angle}\aarg{shear}\marg{semiaxis-h}\%
+\qquad \marg{semiaxis-v}\meta{$\star$}\oarg{settings1}\oarg{settings2}
+\cs{Xellisse}\meta{$\star$}\parg{center}\oarg{angle}\marg{semiaxis-h}\%
+\qquad\marg{semiaxis-v}\oarg{settings1}\marg{settings2}
+\cs{EllipseWithFocus}\meta{$\star$}\parg{vertex1}\parg{vertex2}\parg{vertex3}\parg{focus}
+\cs{SteinerEllipse}\meta{$\star$}\parg{vertex1}\parg{vertex2}\parg{vertex3}\oarg{diameter}
+\end{ttsintassi}
+ All require the semi axis lengths; the \meta{semiaxis-h} and
+ \meta{semiaxis-v} refer to the semi axes before possible rotation by
+ \meta{angle} degrees, and do not make assumptions on which axis is the
+ larger one. The optional parameter \meta{shear} is the angle in degrees
+ by which the vertical coordinate lines are rotated by effect of shearing. 
+ If \meta{shear}, that by default equals zero, is not set
+ to another value, the asterisks of command \cs{Sellisse} and
+ \cs{XSellisse} do not have any effect. Otherwise the asterisk of
+ \cs{Sellisse} forces to draw the ellipse bounding box (rectangle before
+ shearing, parallelogram after shearing) as shown together with some
+ marked special points (the vertices, spline nodes and control points
+ of the quarter circles or quarter ellipses) in figure~\ref{fig:shear}.
+ For \cs{ellipse} the asterisk implies filling, instead of stroking the
+ ellipse contour.
+ The \meta{setting}~1 and~2 refer to the color filling and/or border
+ color, and contour thickness, as already explained. For the
+ \cs{EllipseWithFocus}, the \meta{focus} contains the coordinates
+ of one of the two ellipse foci; such coordinates should point to
+ some position \emph{inside} the triangle. The \cs{SteinerEllipse}
+ requires less data, in the sense that such ellipse is unique; it
+ is the ellipse internally tangent to the triangle at its side middle
+ points.
+
+\end{enumerate}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Examples}\label{sec:examples}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ Here we can show some examples of the advanced \pack{curve2e} commands
+ and of what can be done with this \pack{euclideangeometry} extension.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Straight and curved vectors}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ 
+ Figure~\ref{fig:vectors} shows some vectors and vector arcs with the
+ code used to draw them; as usual some points are described with cartesian
+ coordinates and some with polar ones.
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{8.5}](0.45)
+\unitlength=0.01\linewidth
+\begin{picture}(100,60)
+\GraphGrid(100,60)
+\put(0,30){\vector(1,2){10}}
+\put(20,30){\Vector(10,20)}
+\VECTOR(40,30)(50,50)
+\VVECTOR(60,30)(70,60)
+\Arc(100,60)(80,60){90}
+\VectorArc(0,0)(20,0){90}
+\VectorARC(100,0)(80,0){-90}
+\polyvector(30,0)(35,10)(55,20)(60,0)
+\end{picture}
+\end{Esempio}
+\caption{Some vectors and vector arcs}\label{fig:vectors}
+\end{figure}
+
+
+\begin{figure*}[!tb]
+\begin{Esempio}*[\setfontsize{10}]\centering
+\unitlength=0.006\linewidth\begin{picture}(120,90)
+%
+\RegPolygon(9,20){20}{6}<\linethickness{3pt}\color{red}> 
+\RegPolygon(55,20){20}{7}[90]
+\RegPolygon(100,20){20}{8}[22.5]<\linethickness{0.5ex}\color{blue}>
+%
+\put(0,50){%
+  \RegPolygon(9,20){20}{3}\RegPolygon(9,20){20}{3}[30]
+  \RegPolygon(9,20){20}{3}[60]\RegPolygon(9,20){20}{3}[90]
+%
+  \RegPolygon*(55,20){20}{4}<\color{green}>
+  \RegPolygon(55,20){20}{4}<\linethickness{1ex}>
+%
+  \RegPolygon*(100,20){20}{4}[45]<\color{orange}>
+  \RegPolygon(100,20){20}{4}[45]<\linethickness{1ex}\color{blue}>
+}
+\end{picture}
+\end{Esempio}
+\caption{A variety of polygons and their codes}\label{fig:variety-of-polygons}
+\end{figure*}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Polygons}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ Figures~\ref{fig:pentagon} and~\ref{fig:filled-pentagon} display
+ a normal and a color filled pentagon with their codes.
+ Figure~\ref{fig:variety-of-polygons} shows a variety of polygons
+ with their codes.
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{10}](0.8)
+\unitlength=0.5mm
+\begin{picture}(40,32)(-20,-17)
+\polyline(90:20)(162:20)(234:20)(306:20)(378:20)(90:20)
+\end{picture}
+\end{Esempio}
+\caption{A normal polygon drawn with \cs{polyline}}
+\label{fig:pentagon}
+\end{figure}
+
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{10}](0.8)
+\unitlength=.5mm
+\begin{picture}(40,32)(-20,-20)
+\color{magenta}
+\polygon*(90:20)(162:20)(234:20)(306:20)(378:20)
+\end{picture}
+\end{Esempio}
+\caption{A filled polygon drawn with \cs{polygon}}
+\label{fig:filled-pentagon}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \subsection{Dashed and dotted lines}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ For dotted lines there is a possibility of specifying the dot size;
+ it can be specified with an explicit unit of measure, or, if no unit is
+ specified, it is assumed to be “points”. The \cs{Dotline} takes care
+ of transforming the implied or the explicit dimension in multiples of
+ \cs{unitlength}. Figure~\ref{fig:DashDot} shows some examples with their
+ codes.
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{10}](0.65)
+\unitlength=1mm
+\begin{picture}(40,40)
+\GraphGrid(40,40)
+\Dashline(0,0)(40,10){4}
+\put(0,0){\circle*{2}}
+\Dashline(40,10)(0,25){4}
+\put(40,10){\circle*{2}}
+\Dashline(0,25)(20,40){4}
+\put(0,25){\circle*{2}}
+\put(20,40){\circle*{2}}
+\Dotline(0,0)(40,40){2}[0.75mm]
+\put(40,40){\circle*{2}}
+\end{picture}
+\end{Esempio}
+\caption{Dashed and dotted lines}\label{fig:DashDot}
+\end{figure}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Generic curves}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ With the \cs{Curve} macro it is possible to make line art or filled shapes. Figures~\ref{fig:hearts} show the same shape, the first just stroked and the second color filled. 
+
+\begin{figure}[!htp]
+\begin{Esempio}[\setfontsize{9}](0.65)
+\unitlength=8mm\relax
+\begin{picture}(5,5)
+\put(0,0){\framebox(5,5){}}\thicklines\roundcap
+\Curve(2.5,0)<0.1,1>(5,3.5)<0,1>%
+  (4,5)<-1,0>(2.5,3.5)<-0.1,-1.2>[-0.1,1.2]%
+  (1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<0.1,-1>
+\end{picture}
+\end{Esempio}
+
+\vspace*{2\baselineskip}
+
+\begin{Esempio}[\setfontsize{9}](0.65)
+\unitlength=8mm\relax
+\begin{picture}(5,5)
+\put(0,0){\framebox(5,5){}}\thicklines\roundcap
+\color{orange}\relax
+\Curve*(2.5,0)<0.1,1>(5,3.5)<0,1>%
+(4,5)<-1,0>(2.5,3.5)<-0.1,-1.2>[-0.1,1.2]%
+(1,5)<-1,0>(0,3.5)<0,-1>(2.5,0)<0.1,-1>
+\end{picture}
+\end{Esempio}
+\caption{A stroked and a filled heart shaped contour}
+\label{fig:hearts}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{The \cs{multiput} command}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ The new \cs{multiput} and \cs{xmultiput} commands are extensions of the 
+ original \cs{multiput} macro; both are used to put a number of objects 
+ according to a discrete law; but they can produce surprising effects. 
+ Figure~\ref{fig:multiput} displays several examples. As it possible 
+ to see, the black dots are evenly distributed along the canvas diagonal; 
+ the green filled squares are along a sloping down line inclined by 
+ $15^\circ$ as specified by the polar coordinates of the \meta{increment}; 
+ the blue filled triangles are distributed along a parabola; the red 
+ stroked diamonds are distributed along a half sine wave.
+
+\noindent\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{8}](0.55)
+\unitlength=0.01\linewidth
+\begin{picture}(100,100)
+\GraphGrid(100,100)
+\multiput(0,0)(10,10){11}{\circle*{2}}
+\color{blue!70!white}
+\multiput(0,0)(10,0){11}{%
+\RegPolygon*{2}{3}<\color{blue!70!white}>}%
+  [\GetCoord(\R)\X\Y
+  \edef\X{\fpeval{\X+10}}
+  \edef\Y{\fpeval{(\X/10)**2}}
+  \CopyVect\X,\Y to\R]
+\multiput(0,0)(10,1){11}{%
+\RegPolygon{2}{4}<\color{magenta}>}%
+  [\GetCoord(\R)\X\Y
+   \edef\X{\fpeval{\X+10}}
+   \edef\Y{\fpeval{sind(\X*1.8)*100}}
+   \CopyVect\X,\Y to\R]
+\multiput(50,50)(-15:5){11}{%
+\RegPolygon*{2}{4}[45]<\color{green!60!black}>}
+\end{picture}
+\end{Esempio}
+\caption{Some examples of the \meta{handler} optional argument}
+\label{fig:multiput}
+\end{figure}
+
+ Another interesting construction is a clock quadrant; this is shown
+ in figure~\ref{fig:orologio}
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{9.5}](0.525)
+\unitlength=0.0095\linewidth
+\begin{picture}(100,100)
+\GraphGrid(100,100)
+\put(50,50){\thicklines\circle{100}}
+\xmultiput[50,50](60:35)(-30:1){12}%
+  {\makebox(0,0){\circle*{2}}}%
+    [\MultVect\R by\D to\R]% 
+\xmultiput[50,50](60:40)(-30:1){12}%
+  {\ArgOfVect\R to\Ang
+    \rotatebox{\fpeval{\Ang-90}}%
+    {\makebox(0,0)[b]{%
+       \Roman{multicnt}}}}%
+      [\Multvect{\R}{\D}\R]
+\thicklines\put(50,50){\circle*{4}}
+\put(50,50){\Vector(37.5:30)}
+\put(50,50){\Vector(180:33)}
+\end{picture}
+\end{Esempio}
+\caption{Usage example of the \texttt{\string\xmultiput} command}
+\label{fig:orologio}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \subsection{Drawing mathematical functions}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ Figure~\ref{fig:iperbole} shows an equilateral hyperbola; since it
+ has asymptotes, the drawing must be carefully done avoiding overflows,
+ parts of drawing out of the \env{picture} area. Nevertheless
+ the possibility of describing mathematical functions in terms of L3
+ functions (in spite of the same name, they are completely different
+ things) makes it possible to exploit the \meta{settings} argument to
+ do the job with \cs{xmultiput}.
+
+\begin{figure*}[!tp]
+\begin{Esempio}[\setfontsize{9}](0.5)%
+\unitlength=0.008\linewidth
+\begin{picture}(100,100)
+\GraphGrid(100,100)
+\VECTOR(0,0)(100,0)\Pbox(100,0)[tr]{x}[0]
+\VECTOR(0,0)(0,100)\Pbox(0,100)[tr]{y}[0]
+\Pbox(0,0)[r]{O}[3pt]
+\thicklines
+\moveto(10,100)\countdef\I=2560 \I=11
+\xmultiput(0,0)(1,0){101}%
+  {\lineto(\I,\fpeval{1000/\I})}%
+  [\advance\I by1 \value{multicnt}=\I]
+\strokepath
+\end{picture}
+\end{Esempio}
+\caption{An equilateral hyperbola drawn with a thinly sampled piecewise
+ continuous line}\label{fig:iperbole}
+\end{figure*}
+
+ A more complicated drawing can be done by expressing the function to draw
+ with parametric equations; the idea is to code the math formulas
+\[\begin{cases}
+x(t) = f_1(t)\\
+y(t) = f_2(t)
+\end{cases}\]
+because it is easy to code the $x$ and the $y$ component and use the
+ \cs{fpdowhile} command to trace the curve with a piecewise continuous
+ line; actually a continuous line with a piecewise continuous derivative;
+ it is important to sample the curve in a sufficient dense way.
+ A heart shaped mathematical function taken from the internet
+ \footnote{\url{http://mathworld.wolfram.com/HeartCurve.html} reports 
+ several formulas, including the cardioid, but the one we use here is 
+ a different function} is the following
+\begin{align*}
+x(t) &= \sin^3(t)\\
+y(t) &= \dfrac{13\cos(t)-5\cos(2t)-2\cos(3t)-\cos(4t)}{16}
+\end{align*}
+ Figure~\ref{fig:math-heart} displays the graph, and its code, and, most
+ important, the L3 definition of the parametric equations. Compared to
+ the previous equations we applied a scale factor and added the final
+ term \texttt(2.4) in order to shift a little bit the drawing so as to
+ vertically center it . 
+
+\begin{figure*}[!tbp]
+\begin{minipage}{\textwidth}
+\def\codefraction{0.5}
+\dimendef\Wuno=2000 \Wuno=\codefraction\textwidth
+\dimendef\Wdue=2002 \Wdue=\dimexpr\textwidth-\columnsep-\Wuno\relax
+\begin{minipage}{\Wuno}\setfontsize{6.9}%
+\begin{verbatim}
+%              Parametric equations of the "heart"
+\providecommand\heart[3]{%
+\edef\X{\fpeval{#1*16*(sind(#2)^3)}}
+\edef\Y{\fpeval{#1*(13*cosd(#2) - 5*cosd(2*#2) 
+  - 2*cosd(3*#2) -cosd(4*#2)+2.4)}}
+\CopyVect\X,\Y to#3}
+%
+%                                         Drawing
+\unitlength=0,005\linewidth
+\begin{picture}(200,200)(-100,-100)
+\GraphGrid(-100,-100)(200,200)
+\VECTOR(-100,0)(100,0)\Pbox(100,0)[tr]{x}[0]
+\VECTOR(0,-100)(0,100)\Pbox(0,100)[tl]{y}[0]
+\Pbox(0,0)[tr]{O}\linethickness{1pt}\bgroup
+\edef\scala{\fpeval{100/16}}
+\countdef\I=2560 \I=0\roundjoin
+\fpdowhile{\I !>360}{\heart\scala\I\Punto
+\ifnum\I=0 \moveto(\Punto)\else \lineto(\Punto)\fi
+\advance\I by 3}\strokepath\egroup
+\end{picture}\end{verbatim}
+\end{minipage}
+\hfill
+\begin{minipage}{\Wdue}\raggedleft
+\setfontsize{8}%
+%
+\providecommand\heart[3]{%
+\edef\X{\fpeval{#1*16*(sind(#2)^3)}}
+\edef\Y{\fpeval{#1*(13*cosd(#2) - 5*cosd(2*#2) 
+  - 2*cosd(3*#2) -cosd(4*#2)+2.4)}}
+\CopyVect\X,\Y to#3}
+%
+\unitlength=0,005\linewidth
+\begin{picture}(200,200)(-100,-100)
+\GraphGrid(-100,-100)(200,200)
+\VECTOR(-100,0)(100,0)\Pbox(100,0)[tr]{x}[0]
+\VECTOR(0,-100)(0,100)\Pbox(0,100)[tl]{y}[0]
+\Pbox(0,0)[tr]{O}\linethickness{1pt}\bgroup
+\edef\scala{\fpeval{100/16}}
+\countdef\I=2560 \I=0\roundjoin
+\fpdowhile{\I !>360}{\heart\scala\I\Punto
+\ifnum\I=0 \moveto(\Punto)\else \lineto(\Punto)\fi
+\advance\I by 3}\strokepath\egroup
+\end{picture}
+\end{minipage}
+\end{minipage}
+\caption{A heart shaped mathematical function drawn with a thinly sampled
+ piecewise continuous line}\label{fig:math-heart}
+\end{figure*}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Triangles and their special lines}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ Triangles have special lines; they are the median, the height, and the
+ bisector lines. They join each vertex with a specific point of the
+ apposite side, respectively with the middle point, the intersection 
+ with the side perpendicular line, and the intersection with the bisector 
+ line. Figure~\ref{fig:MHBlines}  displays the construction of the three 
+ special lines relative to a specific vertex. Thanks to the macros 
+ described earlier in this list, this drawing is particularly simple; most
+ of the code is dedicated to labelling the various points and to
+ assign coordinate values to the macros that are going to use them
+ in a symbolic way. The generic triangle (not a regular polygon) requires 
+ one line, and the determination of the intersections of the lines with
+ the suitable triangle side, and their tracing requires two code lines
+ each.
+
+\begin{figure}[!tb]\centering
+\begin{Esempio}[\setfontsize{6.9}](0.4)
+\unitlength=0.008\linewidth
+\begin{picture}(100,100)(0,-10)
+\GraphGrid(0,-10)(100,100)
+\def\Puno{0,0} \def\Pdue{0,80} \def\Ptre{100,60}
+ {\thicklines\polygon(\Puno)(\Pdue)(\Ptre)}%
+ \Pbox(\Puno)[tc]{P_1}[1]
+\Pbox(\Pdue)[bc]{P_2}[1]\Pbox(\Ptre)[bc]{P_3}[1]
+ % Median
+ \TriangleMedianBase\Puno on \Pdue and \Ptre to\M
+ \Pbox(\M)[bc]{M}[1.5]\segment(\Puno)(\M)
+% Height
+ \TriangleHeightBase\Puno on \Pdue and\Ptre to\H
+ \Dotline(\Puno)(\H){2}[1.5]\Pbox(\H)[bc]{H}[1.5]
+% Bisector
+ \TriangleBisectorBase\Puno on\Pdue and\Ptre to\B
+ \Dashline(\Puno)(\B){3}\Pbox(\B)[b]{B}[1.5]
+\end{picture}
+\end{Esempio}
+\caption{A triangle with the median, the height, and the bisector lines
+ from a specific vertex}\label{fig:MHBlines}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \subsection{Special triangle centers}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ Each triplet of a triangle special lines of the same kind intersect each
+ other in a special point; the median lines intersect in the
+ \emph{barycenter}, the height lines in the \emph{orthocenter}, the 
+ bisectors lines in the \emph{incenter}; these centers may be those of 
+ special circles: Figures~\ref{fig:barycenter} to~\ref{fig:circumcenter}; 
+ the \emph{incircle}, centered in the incenter,
+ has a special name, because it has the property of being tangent to all
+ the three triangle sides; there is also the circumcircle that passes
+ through the three vertices, its center is the intersection of the
+ three side axes. There is also the \emph{nine point circle}.
+ Figures~\ref{fig:barycenter}, \ref{fig:orthocenter}, \ref{fig:incenter},
+ and~\ref{fig:circumcenter} display the necessary constructions and,
+ possibly, also the special circles they are centers of.
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{6.5}](0.45)
+\unitlength=0.008\linewidth
+\begin{picture}(100,100)(0,-10)
+\GraphGrid(0,-10)(100,100)
+\def\Puno{0,0}\def\Pdue{0,80}\def\Ptre{100,60}
+{\linethickness{0.6pt}\polygon(\Puno)(\Pdue)(\Ptre)}%
+\Pbox(\Puno)[tl]{P_1}[1.5]%
+\Pbox(\Pdue)[bl]{P_2}[1.5]\Pbox(\Ptre)[bc]{P_3}[1.5]
+ \TriangleMedianBase\Puno on\Pdue and \Ptre to\Mu
+ \TriangleMedianBase\Pdue on\Ptre and \Puno to\Md
+ \TriangleMedianBase\Ptre on\Puno and \Pdue to\Mt
+\Dotline(\Puno)(\Mu){3}[1.5]
+\Dotline(\Pdue)(\Md){3}[1.5]
+\Dotline(\Ptre)(\Mt){3}[1.5]
+\IntersectionOfSegments(\Puno)(\Mu)and(\Pdue)(\Md)to\C
+\Pbox(\C)[t]{B}[2]
+\end{picture}
+\end{Esempio}
+\caption{Determination of the barycenter}\label{fig:barycenter}
+\end{figure}
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{6.5}](0.45)
+\unitlength=0.008\linewidth
+\begin{picture}(100,100)(0,-10)
+\GraphGrid(0,-10)(100,100)
+\def\Puno{0,0}\def\Pdue{0,80}\def\Ptre{100,60}
+{\linethickness{0.6pt}\polygon(\Puno)(\Pdue)(\Ptre)}%
+\Pbox(\Puno)[tl]{P_1}[1.5]%
+\Pbox(\Pdue)[bl]{P_2}[1.5]\Pbox(\Ptre)[bc]{P_3}[1.5]
+ \TriangleHeightBase\Puno on\Pdue and \Ptre to\Hu
+ \TriangleHeightBase\Pdue on\Ptre and \Puno to\Hd
+ \TriangleHeightBase\Ptre on\Puno and \Pdue to\Ht
+\Dotline(\Puno)(\Hu){3}[1.5]
+\Dotline(\Pdue)(\Hd){3}[1.5]
+\Dotline(\Ptre)(\Ht){3}[1.5]
+\IntersectionOfSegments(\Puno)(\Hu)and(\Pdue)(\Hd)to\C
+\Pbox(\C)[t]{H}[2]
+\end{picture}
+\end{Esempio}
+\caption{Determination of the orthocenter}\label{fig:orthocenter}
+\end{figure}
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{6}](0.45)
+\unitlength=0.008\linewidth
+\begin{picture}(100,100)(0,-10)
+\GraphGrid(0,-10)(100,100)
+\def\Puno{0,0}\def\Pdue{0,80}\def\Ptre{100,60}
+{\linethickness{0.6pt}%
+  \polygon(\Puno)(\Pdue)(\Ptre)}%
+\Pbox(\Puno)[tl]{P_1}[1.5]%
+\Pbox(\Pdue)[bl]{P_2}[1.5]
+\Pbox(\Ptre)[bc]{P_3}[1.5]
+ \TriangleBisectorBase\Puno on\Pdue and \Ptre to\Iu
+ \TriangleBisectorBase\Pdue on\Ptre and \Puno to\Id
+ \TriangleBisectorBase\Ptre on\Puno and \Pdue to\It
+\Dotline(\Puno)(\Iu){3}[1.5]
+\Dotline(\Pdue)(\Id){3}[1.5]
+\Dotline(\Ptre)(\It){3}[1.5]
+\IntersectionOfSegments(\Puno)(\Iu)%
+  and(\Pdue)(\Id)to\C
+\Pbox(\C)[t]{I}[2]
+\DistanceOfPoint\C from(\Puno)(\Pdue)to\R
+\Circlewithcenter\C radius\R
+\end{picture}
+\end{Esempio}
+\caption{Determination of the incenter and of the incircle}\label{fig:incenter}
+\end{figure}
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{6.25}](0.45)
+\unitlength=0.01\linewidth
+\begin{picture}(100,110)
+\GraphGrid(100,110)
+\CopyVect20,10to\Pu \Pbox(\Pu)[t]{P_1}
+\CopyVect10,90to\Pd \Pbox(\Pd)[br]{P_2}
+\CopyVect100,70to\Pt \Pbox(\Pt)[l]{P_3}
+{\linethickness{0.6pt}\polygon(\Pu)(\Pd)(\Pt)}%
+\AxisOf\Pd and\Pu to\Mu\Du 
+\AxisOf\Pu and\Pt to\Md\Dd
+\AxisOf\Pt and\Pd to\Mt\Dt
+\IntersectionOfLines(\Mu)(\Du)and(\Md)(\Dd)to\C
+\AddVect\Mu and\Du to\Du\Dotline(\Mu)(\Du){3}[2]
+\AddVect\Md and\Dd to\Dd\Dotline(\Md)(\Dd){3}[2]
+\AddVect\Mt and\Dt to\Dt\Dotline(\Mt)(\Dt){3}[2]
+\Pbox(\C)[t]{C}[2.5]
+\ThreePointCircle*(\Pu)(\Pd)(\Pt)
+\end{picture}\end{Esempio}
+\caption{Determination of the circumcenter and of the circumcircle}\label{fig:circumcenter}
+\end{figure}
+
+ Although these examples require some new simple macros, described
+ in the previous sections; some more more examples can be made that require 
+ more complex macros. Even these macros are just examples. For other
+ applications it is probably necessary to add more macros.
+
+ Let us proceed with the construction of the Steiner ellipse: given a
+ triangle, there exists only one ellipse that is internally tangent to
+ the side middle points.
+
+ The geometrical construction goes on this way; suppose you have to draw
+ the Steiner ellipse of triangle $T$; finding the side middle points
+ has already been shown, but the process to build the ellipse is still
+ to be found. So let us chose a side to work as the base of triangle $T$,
+ and perform an affine shear transformation parallel to the base 
+ so as to move the vertex of triangle $T$, opposite to the base, to the
+ base axis, we get another triangle $T_1$ that is isosceles; if it is
+ not yet so, let us make another compression/expansion affine
+ transformation, so as to get an equilateral triangle $T_2$; this last
+ triangle is particularly simple to handle, because its Steiner ellipse
+ reduces to its incircle. If we apply in reverse order the above
+ transformations we get the Steiner ellipse we were looking for.
+ The only difficult part is the affine shear transformation. 
+
+ The L3 functions we already created take care of all such
+ transformations, but with an optional asterisk we can draw the
+ intermediate passages where triangles $T_2$ and $T_1$ have their base
+ shifted and rotated to be horizontal, so that some translations and
+ rotations are also necessary. Figure~\ref{fig:Steiner-finale} displays
+ the final result and the code necessary to build it.
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{8}](0.5)
+\unitlength=0.01\linewidth
+\begin{picture}(100,110)
+\GraphGrid(100,110)%
+\SteinerEllipse(10,10)(90,20)(60,105)[2]
+\end{picture}\end{Esempio}
+\caption{The Steiner ellipse of a given triangle}
+\label{fig:Steiner-finale}
+\end{figure}
+
+ With just the addition of an asterisk we can draw the whole geometrical
+ construction; see figure~\ref{fig:Steiner-intermedio}
+
+\begin{figure}[!htb]
+\begin{Esempio}[\setfontsize{8}](0.5)
+\unitlength=0.01\linewidth
+\begin{picture}(100,110)(0,-10)
+\GraphGrid(0,-10)(100,110)%
+\SteinerEllipse*(10,10)(90,20)(60,105)[2]
+\end{picture}\end{Esempio}
+\caption{The construction of the Steiner ellipse of a given triangle}
+\label{fig:Steiner-intermedio}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{A triangle internally tangent ellipse given one of its foci}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ It is possible to draw an ellipse that is internally tangent to
+ a triangle if one of its foci is specified; without this specification
+ the problem is not definite, and the number o such ellipses is countless.
+ But with the focus specification, just one ellipse exists with that
+ tangency constraint. It suffices to find the other focus and at least
+ one point of tangency, because the focal distance and the sum of
+ distances of that tangency point from the foci, is sufficient to
+ determine all the parameters required to draw the ellipse.
+
+\begin{figure}[!tb]\centering
+\begin{Esempio}[\setfontsize{7}](0.375)
+\unitlength=0.0065\linewidth
+\begin{picture}(150,150)(-30,-20)
+\GraphGrid(-30,-20)(150,150)
+\EllipseWithFocus%
+  (10,40)(110,10)(0,110)(20,60)
+\end{picture}\end{Esempio}
+\caption{An ellipse internally tangent to a triangle, given a focus}
+\label{fig:ellisse-interna-finale}
+\end{figure}
+
+\begin{figure}[!tb]
+\begin{Esempio}[\setfontsize{7}](0.375)
+\unitlength0.0065\linewidth
+\begin{picture}(150,150)(-30,-20)
+\GraphGrid(-30,-20)(150,150)
+\EllipseWithFocus*%
+  (10,40)(110,10)(0,110)(20,60)
+\end{picture}\end{Esempio}
+\caption{Construction of the ellipse of figure~\ref{fig:ellisse-interna-finale}}\label{fig:ellisse-interna}
+\end{figure}
+
+
+ The geometrical construction is rather complicated; the steps to follow
+ are the following:
+\begin{itemize}[noitemsep]
+
+\item draw the triangle and the given focus $\mathsf{F}$;
+
+\item Find the symmetrical points $\mathsf G_i$ of this focus with 
+ respect to the sides of the triangle;
+
+\item use these three points $\mathsf G_i$ as the vertices of a triangle
+ with which to draw its circumcircle, actually only its center is of
+ interest, because it represents the second focus $\mathsf F'$; the
+ inter focal distance  $2c$; is just le length of vector $F' -F$;
+
+\item join with segments each symmetrical point $\mathsf G_i$ with the
+ second focus $\mathsf F'$ and find their intersections $\mathsf T_i$
+ with the triangle sides; they represent the tangency points of the
+ ellipse to be drawn;
+
+\item use one of these tangency points to find its distances from the
+ foci; their sum gives the ellipse larger axis length $2a$;
+
+\item equation~\eqref{equ:axes-foci} allows to find the second axis
+ length; the segment that joins the foci has the required inclination
+ of the main axis; therefore all necessary pieces of information to
+ draw the ellipse are known.
+
+\end{itemize}
+ Figures~\ref{fig:ellisse-interna-finale} and~\ref{fig:ellisse-interna}
+ display the construction steps and the final result.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \section{Conclusion}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ We have shown that the \amb{picture} environment, extended with
+ this package \pack{euclideangeometry} (that takes care of loading
+ \pack{curve2e} and \pack{pict2e}) can make important diagrams that
+ certainly were not foreseen by Leslie Lamport when he first wrote the
+ code for the initial \env{picture} environment.
+ 
+ The reader can easily understand that this package is far from being
+ exhaustive for all geometrical problema]s to be solved with ruler and
+ compass; it shows a way to add more commands to approach further problems;
+ if any author, who creates new commands, would like to  contribute more
+ macros to this package, I will be happy to integrate his/her contribution
+ to a new version of this package; depending on the contribution, I would
+ be very happy to add its author name to this  package author list; for
+ simpler contributions each contributor will be duly acknowledged.
+
+ Creating new macros to solve more problems is pleasant and more
+ difficult is the problem, greater is the satisfaction in solving it.
+
+
+\begin{center}
+ Have fun with \LaTeX and its potential  applications!
+\end{center}
+
+\end{document}
\ No newline at end of file
diff --git a/Master/texmf-dist/doc/latex/euclideangeometry/euclideangeometry.pdf b/Master/texmf-dist/doc/latex/euclideangeometry/euclideangeometry.pdf
new file mode 100644
index 00000000000..6246a0b6045
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/euclideangeometry/euclideangeometry.pdf
diff --git a/Master/texmf-dist/source/latex/euclideangeometry/euclideangeometry.dtx b/Master/texmf-dist/source/latex/euclideangeometry/euclideangeometry.dtx
new file mode 100644
index 00000000000..6078fe5cfae
--- /dev/null
+++ b/Master/texmf-dist/source/latex/euclideangeometry/euclideangeometry.dtx
@@ -0,0 +1,1588 @@
+% \iffalse
+% !TEX encoding = UTF-8 Unicode
+%<*internal>
+\begingroup
+\input docstrip.tex
+\keepsilent
+\preamble
+
+  Copyright (C)  2020 Claudio Beccari all rights reserved.
+  License information appended
+  
+\endpreamble
+
+\postamble
+
+Distributable under the LaTeX Project Public License,
+version 1.3c or higher (your choice). The latest version of
+this license is at: http://www.latex-project.org/lppl.txt
+
+This work is "maintained"
+
+This work consists of files:
+1) euclideangeometry.dtx, and the derived files euclideangeometry.sty 
+                          and euclideangeometry.pdf,
+2) euclideangeometry-man.tex and the derived file euclideangeometry-man.pdf,
+3) the auxiliary derived file README.txt 
+
+\endpostamble
+\askforoverwritefalse
+
+\generate{\file{euclideangeometry.sty}{\from{euclideangeometry.dtx}{package}}}
+\generate{\file{README.txt}{\from{euclideangeometry.dtx}{readme}}}
+
+\def\tmpa{plain}
+\ifx\tmpa\fmtname\endgroup\expandafter\bye\fi
+\endgroup
+%</internal>
+% \fi
+%
+% \iffalse
+%<*driver>
+\ProvidesFile{euclideangeometry.dtx}%
+%</driver>
+%<package>\NeedsTeXFormat{LaTeX2e}[2019/01/01]
+%<package>\ProvidesPackage{euclideangeometry}%
+%<readme>File README.txt for package euclideangeometry
+%<*package|readme>
+        [2020-02-02 v.0.1.2 Extension package for curve2e]
+%</package|readme>
+%<*driver>
+\documentclass{ltxdoc}\errorcontextlines=100
+\hfuzz 10pt
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern,textcomp}
+\usepackage{mflogo}
+\usepackage{multicol,amsmath,fancyvrb,graphicx,verbatim}
+\usepackage{etoolbox,enumitem,xspace,xcolor,euclideangeometry}
+\providecommand*\diff{\mathop{}\!\mathrm{d}}
+\renewcommand\meta[1]{{\normalfont\textlangle\textit{#1}\textrangle}}
+\renewcommand\marg[1]{\texttt{\{\meta{#1}\}}}
+\providecommand\Marg{}
+\renewcommand*\Marg[1]{\texttt{\{#1\}}}
+\providecommand\oarg{}
+\renewcommand*\oarg[1]{\texttt{[\meta{#1}]}}
+\providecommand\Oarg{}
+\renewcommand*\Oarg[1]{\texttt{[#1]}}
+\providecommand\aarg{}
+\renewcommand*\aarg[1]{\texttt{<\meta{#1}>}}
+\providecommand\Aarg{}
+\renewcommand*\Aarg[1]{\texttt{<#1>}}
+\providecommand\parg{}
+\renewcommand*\parg[1]{\texttt{(\meta{#1})}}
+\providecommand\Parg{}
+\renewcommand*\Parg[1]{\texttt{(#1)}}
+\providecommand\pack{}
+\renewcommand*\pack[1]{{\textsf{#1}}}
+\providecommand\opt{}
+\renewcommand*\opt[1]{{\normalfont\textsl{#1}}}\let\opz\opt
+\providecommand\env{}
+\renewcommand*\env[1]{{\normalfont\textit{\bfseries#1}}}\let\amb\env
+
+\providecommand\TB{\mbox{\normalfont\slshape TUGboat}\xspace}
+
+\AfterEndPreamble{%
+\apptocmd{\TeX}{{}\xspace}{}{}
+\apptocmd{\LaTeX}{{}\xspace}{}{}}
+
+
+\begin{document}\errorcontextlines=100
+\GetFileInfo{euclideangeometry.dtx}
+\title{The \textsf{euclideangeometry} package}
+\author{Claudio Beccari\\[1ex]\texttt{claudio dot beccari at gmail dot com}}
+\date{Version \fileversion~--~Last revised \filedate.}
+\maketitle
+\columnseprule=0.4pt
+\begin{multicols}{2}
+ \tableofcontents
+ \end{multicols}
+ \DocInput{euclideangeometry.dtx}
+\end{document}
+%</driver>
+% \fi
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section*{Preface}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% This file contains the documented code of \pack{euclideangeometry}.
+% The user manual source file \file{euclideangeometry-man.tex} and the
+% readable document is \file{euclideangeometry.pdf}; it should already be
+% installed with your updated  complete \TeX system installation.
+
+% Please refer to the user manual before using this package.
+% \CheckSum{1557}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%\StopEventually{}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%\iffalse
+%<*package>
+%\fi
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{The code}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \subsection{Checking the date of a sufficiently recent \texttt{curve2e}
+% package}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% This package has been already identified by the commands extracted
+% by the |docstrip| package, during  the |.dtx| file compilation.
+% Therefore we start providing the |\GetFileInfo| in case the package
+% is used outside a “documented \TeX\ source”; we just provide the
+% definition from the |ltxdoc| class code. 
+%    \begin{macrocode}
+\makeatletter
+\providecommand\GetFileInfo{}
+\renewcommand\GetFileInfo[1]{%
+  \def\filename{#1}%
+  \def\@tempb##1 v.##2 ##3\relax##4\relax{%
+    \def\filedate{##1}\def\fileversion{##2}\def\fileinfo{##3}}%
+  \edef\@tempa{\csname ver@#1\endcsname}%
+  \expandafter\@tempb\@tempa\relax? ? \relax\relax}
+%    \end{macrocode}  
+%
+% But in order to make the file date comparison for loading a suitable
+% version of |curve2e| we need to strip away its date dash signs,
+% so we define the |\cleanfiledate| macro that defines a |\FileDate| macro
+% with a purely numerical string, suitable for numerical comparisons;
+% if the fetched class file is so old that the date is not in the ISO
+% format (for example with slashes instead of dashes) the macro fails
+% and the interpreter comparison  algorithm produces \TeX\ error messages.
+%    \begin{macrocode}
+\def\cleanfiledate#1-#2-#3!{\unless\ifx#3\empty
+\edef\FileDate{#1#2#3}\else\unless\ifx#2\empty
+\edef\FileDate{#1#2}\else\edef\FileDate{#1}\fi\fi}
+%    \end{macrocode}  
+% In any case, if the test checks that the |curve2e| file date is too old;
+% it warns the user with an emphasised error message on the console,
+% loading this |euclideangeometry| package is stopped and the whole
+% job aborts. The emphasised error message appears like this:
+%\begin{flushleft}\ttfamily\obeylines
+%*************************************************************
+%Package curve2e too old
+%Be sure that your TeX installation is complete and up to date
+%*************************************************************
+%Input of euclideangeometry is stopped and job aborted
+%*************************************************************
+%\end{flushleft}
+% This message should be sufficiently strong in order to avoid using
+% this package with a vintage version of \TeXLive or Mik\TeX. 
+%    \begin{macrocode}
+\RequirePackage{curve2e}
+\GetFileInfo{curve2e.sty}
+\expandafter\cleanfiledate\filedate!
+
+\ifnum\FileDate<20200118 
+\typeout{*************************************************************}
+\typeout{Package curve2e too old}
+\typeout{Be sure that your TeX installation is complete and up to date}
+\typeout{*************************************************************}
+\typeout{Input of euclideangeometry stopped and job aborted}
+\typeout{*************************************************************}
+\expandafter\@@end\fi
+
+%    \end{macrocode}  
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \subsection{Service macros}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% The following macros are useful for typesetting the manual; but they
+% might be useful also for the final user; they mostly define commands
+% that classify the various entities of the programming language with
+% (generally different) font families, series and shapes; but the commands
+% that receive the “name of an argument” enclose this name between math
+% angle brackets.
+%    \begin{macrocode}
+\providecommand\file{}
+\renewcommand*\file[1]{{\normalfont\texttt{#1}}}
+\providecommand\prog{}
+\renewcommand*\prog[1]{{\normalfont\texttt{#1}}}
+\newcommand*\conta[1]{\texttt{\slshape#1}}
+\providecommand\cs{}
+\renewcommand\cs[1]{{\normalfont\texttt{\char92#1}}}
+\providecommand\meta{}
+\renewcommand*\meta[1]{{\normalfont$\langle$\textit{#1}$\rangle$}}
+\providecommand\marg{}
+\renewcommand*\marg[1]{{\ttfamily \char123\meta{#1}\char125}}
+\newcommand*\Marg[1]{{\normalfont\ttfamily\{#1\}}}\let\Arg\Marg
+\providecommand\oarg{}
+\renewcommand*\oarg[1]{{\ttfamily[\meta{#1}]}}
+\newcommand*\Oarg[1]{{\ttfamily[#1]}}
+\newcommand*\aarg[1]{{\ttfamily{<\meta{#1}>}}}
+\newcommand*\Aarg[1]{{\ttfamily<#1>}}
+\newcommand*\Bambiente[1]{\texttt{\char92begin\{#1\}}}
+\newcommand*\Eambiente[1]{\texttt{\char92end\{#1\}}}
+\newcommand*\garg[1]{\texttt{(\splitgarg#1,!)}}\let\parg\garg
+\newcommand*\Garg[1]{\texttt{(#1)}}\let\Parg\Garg
+  \def\secgarg#1,{#1}
+  \def\splitgarg#1,#2!{%
+     \ifstrequal{#2}{}{\meta{#1}}{\meta{#1},\meta{\secgarg#2}}}
+\providecommand\pack{}
+\renewcommand*\pack[1]{{\normalfont\textsf{#1}}}
+\providecommand\opt{}
+\renewcommand*\opt[1]{{\normalfont\textsl{#1}}}\let\opz\opt
+\providecommand\env{}
+\renewcommand*\env[1]{{\normalfont\textit{\bfseries#1}}}\let\amb\env
+%    \end{macrocode}
+% The next commands  are defined in the |babel-italian| module for |babel|;
+% |\ped| introduces a subscript in math mode, but the command may be used
+% also in text mode; the same holds true also for |\ap| that introduces
+% a superscript. |\unit| is a shorthand for attaching the units of measure
+% in math upright roman font, with an unbreakable space between the measure
+% and its units; |\sigla| is the Italian equivalent of |\acronym|, that
+% avoids any conflict with other packages; |\iu| (alias|\gei|) typesets
+% the imaginary unit in roman font according to the ISO regulations;
+% |\eu| defines the roman type “e”, the Neper number, as a math operator;
+% we all know that “e” is not an operator, but it is used very often as
+% the base of an exponential, therefore, in math mode it is better to
+% leave the necessary space before and after it, without any space between
+% it and its exponent. 
+%    \begin{macrocode}
+\providecommand*\ped[1]{\ensuremath{_{\mathrm{#1}}}}
+\providecommand*\ap[1]{\ensuremath{^{\mathrm{#1}}}}
+\providecommand*\unit[1]{\ensuremath{\,\mathrm{#1}}}
+\providecommand*\sigla[1]{\textsc{\lowercase{#1}}}
+\providecommand*\iu{\ensuremath{\mathrm{i}}}\let\gei\iu
+\providecommand*\eu{\ensuremath{\mathop{\mathrm{e}}\nolimits}}
+%    \end{macrocode}
+%
+% The next code is used to typeset the logo of \Ars logo, and the logo
+% of \LyX; the other command |\GuIT| typesets equally well the \GuIT logo;
+% if necessary the |\RequirePackage| macro loads the |xspace| package;
+% therefore these macros do not require any special terminator of the
+% control sequence name, because the |\xspace| command takes care of the
+% necessary space; this command avoids inserting any space if macros are
+% followed by any reasonable sign different form a space, such as
+% punctuation marks, parentheses, quotation marks, and so on.
+%
+%    \begin{macrocode}
+\DeclareRobustCommand*{\Ars}{%
+  \textsf{\lower -.48ex\hbox{\rotatebox{-20}{A}}\kern -.3em{rs}}%
+  \kern -.05em\TeX\unskip\kern -.17em\lower -.357ex\hbox{nica}\xspace}
+  
+\DeclareRobustCommand*\LyX{%
+  L\kern-0.2em\raisebox{-0.5ex}{Y}\kern-0.2em{X}\xspace}
+
+\definecolor{verdeguit}{rgb}{0, 0.40, 0}
+\RequirePackage{iftex}
+\def\GuIT{\mbox{\color{verdeguit}\def\I{\textcolor{black}{I}}%
+\ifPDFTeX
+  \usefont{T1}{lmr}{m}{sc}
+\else
+  \usefont{TU}{lmr}{m}{sc}
+\fi
+g\raisebox{-0.715ex}{\kern-0.26em u}\kern-0.13em\I\kern-0.14em t}\xspace}
+%    \end{macrocode}
+% The following macros |\usecs| and |\whilenum| allow the use the internal
+% \LaTeX control sequence names that contain a |@| sign. They are not so
+% useful in this package, but they might be useful for the end user. 
+%    \begin{macrocode}
+\providecommand*\usecs[1]{\csname#1\endcsname}
+\providecommand*\whilenum[2]{\usecs{@whilenum}#1\do{#2}}
+%    \end{macrocode}
+% Accordingly the following two macros, and their English aliases, allow
+% to delay the output of a (possibly floating) object to a page of the
+% desired parity; they are not infallible, because the the |\afterpage|
+% command defined by package |afterpage| (loaded if necessary) operates
+% on the  typesetting program output routine and the object to be delayed
+% might be lost on the way, or might get delayed too much; these facts
+% take place very seldom, but are not impossible, the solution is to move
+% back several paragraphs the command with is argument.
+%    \begin{macrocode}
+\RequirePackage{afterpage}
+\providecommand\supaginapari[1]{%
+  \afterpage{%
+    \ifodd\value{page}\expandafter\@firstoftwo
+    \else\expandafter\@secondoftwo
+    \fi{\supaginapari{#1}}{#1}
+  }
+}\let\onevenpage\supaginapari
+\providecommand\supaginadispari[1]{%
+  \afterpage{%
+    \ifodd\value{page}\expandafter\@secondoftwo
+    \else\expandafter\@firstoftwo
+    \fi{\supaginadispari{#1}}{#1}
+  }
+}\let\onoddpage\supaginadispari
+%    \end{macrocode}
+%  The following \LaTeX related logos are provided so as to make them
+% coherent with the smart final space we have discussed above. The real
+% difference is that it is necessary to use tue |etoolbox| facilities;
+% therefore the package is (possibly) loaded. With it we add at the end
+% of their original definition the |\xspace| command. But what is the
+% “original definition”? Of course |\TeX| and |\LaTeX| macros are defined
+% within the \LaTeX kernel, but the user might have loaded other packages
+% that have redefined them; therefore these patches are applied only at
+% the preamble end, so if other definitions were provided by other
+% packages loaded in the preamble, such redefinitions are patched.
+% Only |\XeLaTeX| is completely redefined, because the other existing
+% definitions work correctly only with OpenType fonts; with Type~1 fonts
+% and oblique fonts (italics, slanted, etc.) the reversed “E” generally
+% is inclined in the wrong direction; by rotating it, instead of
+% reflecting it, the “E” might not be the best, but al least slants in
+% the right direction.
+%    \begin{macrocode}
+\RequirePackage{etoolbox}
+\AfterEndPreamble{%
+\apptocmd{\TeX}{{}\xspace}{}{}
+\apptocmd{\LaTeX}{{}\xspace}{}{}}
+
+\DeclareRobustCommand*\TeXLive{\TeX\ Live\xspace}
+\DeclareRobustCommand*\MiKTeX{MiK\TeX}
+\DeclareRobustCommand*\MacTeX{Mac\TeX}
+\DeclareRobustCommand*\pdfLaTeX{pdf\/\LaTeX}
+\DeclareRobustCommand*\LuaLaTeX{Lua\LaTeX}
+\DeclareRobustCommand*\XeLaTeX{X\ifdim\fontdimen1\font=0pt\kern-0.15em\fi
+\lower.5ex\hbox{\rotatebox[origin=c]{180}{E}}%
+\ifdim\fontdimen1\font=0pt\kern-0.15em\else\kern-0.275em\fi
+\LaTeX}
+\DeclareRobustCommand*\TikZ{Ti\emph{k}Z\xspace}
+\DeclareRobustCommand*\TUGboat{\textsl{TUG\-boat}\xspace}
+%    \end{macrocode}
+%  The following command is useful to output two column floats when
+% typesetting in |twocolumn| mode.
+%    \begin{macrocode}
+\renewcommand\dbltopfraction{1.}
+%    \end{macrocode}
+% This apparently useless command becomes useful to separate a font
+% selection command or the |\item| command from the following text so
+% as to allow hyphenation in what follows; it is just a very innocent
+% dirty trick, but it works.
+%    \begin{macrocode}
+\def\hz{\hskip0pt}
+%    \end{macrocode}
+% Here we have a command that allows to display some framed code; it is
+% usable also to display the syntax of some commands; and when doing this
+% action the full range of service macros at the beginning of this section
+% play the best of their role. The English aliases for the opening and
+% closing environments are also provided. Attention: do not use |\\|
+% commands to specify new lines, unless you want to produce empty/blank
+% lines; within this environment the source code “end of line” characters
+% are not treated as spaces, but are actually executed according to their
+% name.
+%    \begin{macrocode}
+\newenvironment{ttsintassi}{\begin{lrbox}{0}
+\minipage{\dimexpr\linewidth-2\fboxrule-2\fboxsep}\ttfamily\obeylines}%
+{\endminipage\end{lrbox}\center\fbox{\box0}\endcenter}
+\let\ttsyntax\ttsintassi \let\endttsyntax\endttsintassi
+%    \end{macrocode}
+% The following macro probably will migrate to |curve2e|; meanwhile
+% this redefinition is useful in order to save some input and to make
+% a more flexible macro. We used it virtually in every picture we
+% inserted in the user manual of this package; of course the user does
+% not need to have any drawing superimposed onto a red grid. Actually
+% the grid is useful while drawing; when the image is complete, the
+% grid command may be deleted or its line commented out. The purpose
+% of this macro is to avoid the |\put| command to place the grid; and
+% since its reference point in the original definition is arbitrary,
+% the connection with real coordinates is very small and might become
+% confusing or of little help.
+% Now the syntax is the following:
+%\begin{ttsyntax}
+%\cs(GraphGrid)\parg{reference point or dimensions}\parg{overall dimensions}
+%\end{ttsyntax}
+% where the second argument is optional; if it is missing, the first
+% argument contains the \meta{overall dimensions}, otherwise it
+% contains the lower left corner coordinate that represent the
+% \meta{reference point} that an internal |\put| command will use
+% to place the grid; if no \meta{reference point} is specified, the
+% reference point is \texttt{0,0}, and the grid is |\put| with its
+% lower left corner in the origin of the |picture| coordinates.
+%    \begin{macrocode}
+
+\unless\ifcsname Gr@phGrid\endcsname
+\let\originalGraphGrid\GraphGrid
+\RenewDocumentCommand\GraphGrid{r() d()}{%
+\IfValueTF{#2}{\put(#1){\originalGraphGrid(#2)}}%
+              {\put(0,0){\originalGraphGrid(#1)}}}\fi
+%    \end{macrocode}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%\subsection{Labelling}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% While doing any graphical geometrical drawing it is necessary to label
+% points, lines, angles and other such items. Non measurable labels should
+% be in upright sans serif font, according to the ISO regulations, but here
+% we are dealing witt point identified by macros the contain their
+% (cartesian or polar) coordinates that very often are both labels
+% and math variables.
+%
+% Here we provide a versatile macro that can do several things. Its name
+% is |\Pbox| and it produces a box containing the label in math format. By
+% default the point label is typeset with the math font variant produced
+% by command |\mathsf|, but the macro is sufficiently versatile to allow
+% other settings; It accepts several optional arguments, therefore it syntax is particular:
+%\begin{ttsyntax}
+%\cs{Pbox}\parg{coordinates}\oarg{alignment}\marg{label}\oarg{diameter}\meta{$\star$}\aarg{angle}
+%\end{ttsyntax}
+% where \meta{coordinates} are the coordinates where to possibly set a
+% black dot with the specified \meta{diameter}; in any case it is the
+% reference point of the \meta{label}; the \meta{alignment} is formed by
+% the usual letters \texttt{t, b, c, l, r} that can be paired in a
+% coherent way (for example the couple \texttt{tb} is evidently
+% incoherent, as well as \texttt{lr}), but in absence of this optional
+% specification, the couple \texttt{cc} is assumed; most often than
+% not, the label position becomes such that when the user reviews the
+% document drafts, s/he understands immediately that s/he forgot to specify
+% some reasonable \meta{alignment} codes. Think of the \meta{alignment}
+% letters as the position of the reference point with respect to the
+% the \meta{label} optical center. The optional \meta{angle} argument
+% produces a rotation of the whole label by that angle; it may be used
+% in several circumstances, especially when the label is just text, to
+% produce, for example,  a sideways legend. It is useful also when the
+% labels are produced within a rotated box, in order to counterrotate them.
+% 
+% The optional asterisk draws a frame around the \emph{label}. Notice that
+% the separator between the visible or the invisible frame and the box
+% contents varies according the the fact the the \meta{alignment}
+% specification contains just one or two letter codes; this is useful,
+% because the diagonal position of the label should be optically equal
+% to the gap that exists between the reference point  and the \meta{label}
+% box.
+%
+% If the \meta{diameter} is zero, no dot is drawn, the whole \meta{label}
+% is typeset with the |\mathit| math font; otherwise only the first symbol
+% of a math expression si typeset in sans serif. The presence of subscripts
+% makes the labels appear more distant from their reference point; the same
+% is true when math symbols, even without subscripts, are used, because
+% of the oblique nature of the math letters alphabet.
+% 
+% If some text has to be printed as a label, it suffices to surround it
+% with  dollar signs, that switch back to text mode when the default mode
+% is the math one. With this kind of textual labels it might be convenient
+% to use the optional asterisk to frame the text.
+%    \begin{macrocode}
+\providecommand\Pbox{}
+\RenewDocumentCommand\Pbox{D(){0,0} O{cc} m O{0.5ex} s D<>{0}}{%
+\put(#1){\rotatebox{#6}{\makebox(0,0){%
+\dimendef\Dim=2566\relax
+\settowidth\Dim{#2}%
+\edef\Rapp{\fpeval{\Dim/{1ex}}}%
+\fptest{\Rapp > 1.5}{\fboxsep=0.5ex}{\fboxsep=0.75ex}%
+\IfBooleanTF{#5}{\fboxrule=0.4pt}{\fboxrule=0pt}%
+\fptest{#4 = 0sp}%
+  {\makebox(0,0)[#2]{\fbox{$\relax#3\relax$}}}%
+  {\edef\Diam{\fpeval{(#4)/\unitlength}}%
+   \makebox(0,0){\circle*{\Diam}}%
+   \makebox(0,0)[#2]{\fbox{$\relax\mathsf#3\relax$}}%
+  }}}%
+}\ignorespaces}
+%    \end{macrocode}
+% The following command, to be used always within a group, or a
+% environment or inside a box, works only with piecewise continuously
+% scalable font collection, such as, for example, the Latin Modern
+% fonts, or with continuously scalable fonts, such as, for example,
+% the Times ones. They let the operator select, for the scope of the
+% command ,any size, even fractional so as to fine adjust the text
+% width in the space allowed for it; it is particularly useful with
+% the monospaced fonts, that forbid hyphenation, and therefore cannot
+% be adjusted to the current line width.
+%    \begin{macrocode}
+\DeclareRobustCommand\setfontsize[2][1.2]{%
+  \linespread{#1}\fontsize{#2}{#2}\selectfont}
+%    \end{macrocode}
+% With OpenType fonts there should not be any problems even with math
+% fonts; with Type~1 fonts the only scalable fonts I know of, are the
+% LibertinusMath fonts, usable through the LibertinusT1math package, are
+% also the only ones that have 8~bit encoded math fonts (256
+% glyph fonts), while the standard default Type~1 math fonts are just
+% 7~bit encoded (128 glyphs fonts).
+%
+% Another useful labelling command is |Zbox|; this command is an
+% evolution of a command that I been using for years in several
+% documents of mine. It uses some general text, not necessarily
+% connected to a particular point of the |picture| environment,
+% as a legend; It can draw short text as a simple horizontal box,
+% and longer texts as a vertical box of specified width and height
+%
+% Is syntax is the following:
+%\begin{ttsyntax}
+%\cs{Zbox}\parg{position}\parg(dimensions)\oarg{alignment}\marg{text}
+%\end{ttsyntax}
+% where \meta{position} is where the reference point of the box has
+% to be put in the picture; \meta{dimensions} are optional; if not
+% specified, the  box is a horizontal one, and it is as wide as its
+% contents; if it is specified, it must be a comma separated list
+% of two integer or fractional numbers that are the width and the
+% height of the box; if the height is specified as zero, the width
+% specifies a horizontal box of that width; \meta{alignment} is optional
+% and is formed by one or two coherent letter codes from the usual set
+% \texttt{t, b, c, l, r}; if the \meta{alignment} is absent, the
+% default alignment letters are \texttt{bl}, i.e. the box reference
+% point is the bottom left corner; \meta{text} contains general text,
+% even containing some math.
+%    \begin{macrocode}
+
+\def\EUGsplitArgs(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}
+
+\providecommand\Zbox{}
+\RenewDocumentCommand\Zbox{R(){0,0} D(){0,0} O{bl} m}{%
+\EUGsplitArgs(#2)\ZboxX\ZboxY % separa la x e la y della scatola
+\fboxsep=2\unitlength
+\ifnum\ZboxX=\csuse{z@}
+  \def\ZTesto{\fbox{#4}}%
+\else
+  \ifnum\ZboxY=\csuse{z@}
+    \def\ZTesto{\fbox{\parbox{\ZboxX\unitlength}{#4}}}%
+  \else
+    \def\ZTesto{%
+    \setbox2560=\hbox{\fbox{%
+         \parbox[c][\ZboxY\unitlength][c]{\ZboxX\unitlength}{#4}}}%
+      \dimen2560=\dimexpr(\ht2560 +\dp2560)/2\relax
+      \ht2560=\dimen2560\relax 
+      \dp2560=\dimen2560\relax
+      \box2560%
+      }%
+  \fi
+\fi
+\put(#1){\makebox(0,0)[#3]{\ZTesto}}\ignorespaces}
+%    \end{macrocode}
+
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%\subsection{The \texttt{Example} environment}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% Another very useful facility is the |Esempio| environment;
+% it has an English alias |Example|, but we suggest to use the Italian
+% name, because it is safer in order to avoid conflicts with other packages;
+% this environment is very handy to typeset a stretch of code side by
+% side its typeset result obtained by executing it; it has advantages and
+% disadvantages, though. A similar environment was used to show the
+% examples in the \emph{The \LaTeX Companion}; another similar environment
+% may be used from the |tcolorbox| package (by T.F.~Sturm, read its
+% documentation by using the terminal command |texdoc tcolorbox|)
+% environment |tcolorbox| and using the keyword |sidebyside| among the
+% options; package |fancyvrb| also offers a similar environment.
+% The critical disadvantage is that that you cannot use any of these
+% environments, not even this |Esempio| one, when typesetting a
+% documented \TeX file, such as the one you are reading now. In facts
+% the |.dtx| file has all text lines that start with a comment character,
+% and the code parts, to be output to a |.sty|, or |.cls|, or whatever
+% other file used by the \TeX system to format documents, are explicitly
+% marked with the special delimiters \verb*|%    \begin{macrocode}| and
+% \verb*|%    \end{macrocode}|.
+%
+% Therefore with those packages and with this |Esempio| environment
+% the code you want to show appears with all lines with a |%| at the
+% beginning. This certainly does not help the description of a software
+% and this is the very reason why this package has a separate
+% |euclideangeometry-man| manual typeset with the |article| class. 
+%
+% Nevertheless this environment makes use of the facilities of the
+% |verbatim| environment.
+%
+%    \begin{macrocode}
+\RequirePackage{verbatim}
+\newwrite\example@out
+\ProvideDocumentEnvironment{Esempio}{s O{\normalsize} D(){0.40}}
+{\par\addvspace{3.0ex plus 0.8ex minus 0.5ex}\vskip -\parskip
+\dimendef\Wboxu=2570 \dimendef\Wboxd=2572
+\Wboxu=#3\textwidth\relax
+\Wboxd=\dimexpr\linewidth-\columnsep-\Wboxu\relax
+\begingroup
+\@bsphack
+\immediate\openout\example@out\jobname-temp.tex
+\let\do\@makeother\dospecials\catcode`\^^M\active
+\def\verbatim@processline{%
+  \immediate\write\example@out{\the\verbatim@line}}%
+\verbatim@start\relax}%
+{\immediate\closeout\example@out\@esphack\endgroup
+\begin{minipage}{\textwidth}%
+\IfBooleanTF{#1}{\begin{minipage}{\textwidth}}{\begin{minipage}{\Wboxu}}%
+#2\relax
+\verbatiminput{\jobname-temp.tex}
+\end{minipage}%
+\IfBooleanTF{#1}{\par\bigskip}{\hfill}%
+\IfBooleanTF{#1}{\begin{minipage}{\textwidth}}{\begin{minipage}{\Wboxd}}%
+\raggedleft
+\input{\jobname-temp}
+\end{minipage}
+\end{minipage}\par
+%\medskip
+%\par\addvspace{3.0ex plus 0.8ex minus 0.5ex}\vskip -\parskip
+}
+%    \end{macrocode}
+
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \subsection{Service macros for ellipses}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% The |\ellisse| has a control sequence name in Italian; it differs for
+% just one letter from the name |ellipse| English name, but we cannot use
+% the latter one because it may conflict with other packages loaded by
+% the user; actually this command and the next one are just  shortcuts 
+% for executing more general commands with specific sets of arguments.
+% For details and syntax, please refer yourself to
+% section~\ref{ssec:ellissi}
+%    \begin{macrocode}
+
+\NewDocumentCommand\ellisse{ s m m}{%
+\IfBooleanTF{#1}%
+  {\let\fillstroke\fillpath}%
+  {\let\fillstroke\strokepath}%
+\Sellisse{#2}{#3}%
+}
+
+\NewDocumentCommand\Xellisse{ s D(){0,0} O{0} m m O{} o}{%
+\IfBooleanTF{#1}%
+  {\XSellisse*(#2)[#3]{#4}{#5}[#6][#7]}%
+  {\XSellisse(#2)[#3]{#4}{#5}[#6][#7]}%
+}
+%    \end{macrocode}
+%
+% We do not know if the following macro |\polyvector| may be useful for
+% euclidean geometry constructions, but it may be useful in block
+% diagrams; it is simply a polyline where the last segment is a geometrical
+% vector. As in polyline the number of recursions is done until the last
+% specified coordinate pair; recognising that it is the last one, instead
+% of drawing a segment, the macro draws a vector.
+%
+%    \begin{macrocode}
+
+\def\polyvector(#1){\roundcap\def\EUGpreviouspoint{#1}\EUGpolyvector}
+\def\EUGpolyvector(#1){%
+\@ifnextchar({%
+    \segment(\EUGpreviouspoint)(#1)\def\EUGpreviouspoint{#1}\EUGpolyvector}%
+  {\VECTOR(\EUGpreviouspoint)(#1)}%
+}
+%    \end{macrocode}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \subsection{Processing lines and segments}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% The next macros are functional for the geometric constructions we are
+% going to make: finding the intersection of lines or segments,
+% finding the lengths and arguments of segments, directions, distances,
+% distance of a point from a line or a segment, the symmetrical
+% point of a another one specified with respect to a given center of
+% symmetry; the axes of segments, the solutions of the relationship
+% between the semi axes of an ellipse and the semi focal distance,
+% and so on.
+%
+% Most of these commands have delimited arguments; the delimiters
+% may be the usual parentheses, but they may be keywords; many
+% commands contain  the keyword \texttt{to}, not necessarily the
+% last one; the arguments before such keyword may be entered as
+% ordered comma separated numerical couples, or comma separated
+% macros the containing scalar values; or they may be macros that
+% contain the ordered couples representing vectors or directions;
+% they all may be in cartesian or polar form. Remember that such
+% ordered couples are complex numbers, representable by
+% vectors applied to the origin of the axes; therefore sometimes it is
+% necessary that the underlying commands execute some vector
+% differences so as to work with generic vectors.
+%
+% On the opposite the output values, i.e. the argument after that
+% \texttt{to} keyword, should be tokens that can receive a definition,
+% in general macros, to which the user should assign a mnemonic name;
+% s/he should use such macros for further computations or for drawing
+% commands.
+%
+% The first and principal command is |\IntersectionOfLines| and
+% it has the following syntax:
+%\begin{ttsyntax}
+%\cs{IntersectionOfLines}\parg{point1}\parg{dir1}and\parg{point2}\parg{dir2}to\meta{crossing}
+%\end{ttsyntax}
+% where \meta{point1} and \meta{dir1} are respectively a point of the
+% first line and its \emph{direction}, not a second point, but the
+% \emph{direction} — it is important to stress this point; similarly
+% for the second line; the output is stored in
+% the macro that identifies the \meta{crossing} point. The directions
+% do not need to be expressed with unit vectors, but the lines must not
+% be parallel or anti parallel (equal directions or differing by
+% $180^\circ$); the macro contains a test that checks this anomalous
+% situation because an intersection at infinity or too far away
+% ($2^{14}-1$ typographical points, approximately 5,758\,m) is of no
+% interest; in case, no warning message is issued, the result is
+% put to \texttt{0,0}, and the remaining computations become nonsense.
+% It is a very unusual situation and I never encountered~it;
+% nevertheless\dots
+
+%    \begin{macrocode}
+
+\def\IntersectionOfLines(#1)(#2)and(#3)(#4)to#5{\bgroup
+\def\IntPu{#1}\def\Uu{#2}\def\IntPd{#3}\def\Ud{#4}%
+ \DirOfVect\Uu to\Du 
+ \DirOfVect\Ud to\Dd 
+ \XpartOfVect\Du to \a  \YpartOfVect\Du to \b 
+ \XpartOfVect\Dd to \c  \YpartOfVect\Dd to \d 
+ \XpartOfVect\IntPu to \xu \YpartOfVect\IntPu to \yu 
+ \XpartOfVect\IntPd to \xd \YpartOfVect\IntPd to \yd 
+ \edef\Den{\fpeval{-(\a*\d-\b*\c)}}%
+ \fptest{abs(\Den)<1e-5}{% almost vanishing determinant
+   \def#5{0,0}%
+ }{% Determinant OK
+   \edef\Numx{\fpeval{(\c*(\b*\xu-\a*\yu)-\a*(\d*\xd-\c*\yd))/\Den}}%
+   \edef\Numy{\fpeval{(\d*(\b*\xu-\a*\yu)-\b*(\d*\xd-\c*\yd))/\Den}}%
+   \CopyVect\Numx,\Numy to\Paux
+   \edef\x{\egroup\noexpand\edef\noexpand#5{\Paux}}\x\ignorespaces}}
+%    \end{macrocode}
+%
+% The |IntersectionOfSegments| macro is similar but in input it
+% contains the end points of two segments: internally it uses
+% |\IntersectionOfLines| and to do so it has to determine the
+% directions of both segments. The syntax is the following:
+%\begin{ttsyntax}
+%\cs{IntersectionOfSegments}\parg{point11}\parg{point12}and\parg{point21}\parg{point22}
+%\qquad to\meta{crossing}
+%\end{ttsyntax}
+% The \meta{crossing} point might fall outside one or both segments.
+% It is up to the users to find out if the result is meaningful
+% or nonsense. Two non parallel lines are infinitely long in both
+% directions and any \meta{crossing} point is acceptable; with
+% segments the situation might become nonsense.
+%    \begin{macrocode}
+
+\def\IntersectionOfSegments(#1)(#2)and(#3)(#4)to#5{%
+\SubVect#1from#2to\IoSvectu \DirOfVect\IoSvectu to\DirIoSVecu
+\SubVect#3from#4to\IoSvectd \DirOfVect\IoSvectd to\DirIoSVecd
+\IntersectionOfLines(#1)(\DirIoSVecu)and(#3)(\DirIoSVecd)to#5\ignorespaces}
+%    \end{macrocode}
+%
+% An application of the above intersections is formed by the next
+% two macros; they find the axes of a couple of sides of a triangle
+% and use their base point and direction to identify two lines the
+% intersection of which is the circumcenter; the distance of one
+% base point from the circumcenter is the radius of the circumcircle
+% that can be drawn  with the usual macros. We have to describe
+% the macros |\AxisOf| and |CircleWithCenter| and we will do it in
+% a little while. Meanwhile the syntax of the whole macro is the
+% following:
+%\begin{ttsyntax}
+%\cs{ThreePointCircle}\meta{$\star$}\parg{vetex1}\parg{vertex2}\parg{vertex3}
+%\end{ttsyntax}
+% where the three vertices are the three points where the circle
+% must pass, but they identify also a triangle. Its side axes
+% intersect in one point that by construction is at the same
+% distance from the three vertices, therefore it is the center of
+% the circle that passes through the three vertices. A sub product
+% of the computations is the macro |\C| that contains the center
+% coordinates. If the optional asterisk is used the whole drawing
+% is executed, while if it is missing, only the |\C| macro remains
+% available but the user is responsible to save/copy its value
+% into another macro; for this reason another macro should be more
+% easy to use; its syntax is the following: 
+%\begin{ttsyntax}
+%\cs{ThreePointCircleCenter}\parg{vetex1}\parg{vertex2}\parg{vertex3}
+%\qquad to\meta{center}
+%\end{ttsyntax}
+% where the vertices have the same meaning, but\meta{center} is
+% the user chosen macro that contains the center coordinates.
+%
+%    \begin{macrocode}
+
+\NewDocumentCommand\ThreePointCircle{s d() d() d()}{%
+\AxisOf#2and#3to\Mu\Du \AxisOf#2and#4to\Md\Dd
+\IntersectionOfLines(\Mu)(\Du)and(\Md)(\Dd)to\C
+\SubVect#2from\C to\R 
+\IfBooleanTF{#1}{\CircleWithCenter\C Radius\R}{}\ignorespaces}
+
+\NewDocumentCommand\ThreePointCircleCenter{d() d() d() m}{%
+\ThreePointCircle(#1)(#2)(#3)\CopyVect\C to#4}
+%    \end{macrocode}
+%
+% There are some useful commands that help creating |picture|
+% diagrams in an easier way; for example one of the above described
+% commands internally uses |\CircleWithCenter|. It is well known
+% that the native |picture| command |\circle| requires the
+% specification  of the diameter but many |euclideangeometry|
+% commands already get the distance of two points, or the magnitude
+% of a segment, or similar objects that may be used as a radius, rather
+% than the diameter; why should we not have macros that simultaneously
+% compute the require diameter and draw the circle. Here there are two
+% such macros; they are similar to one another  but their names differ in
+% capitalisation, but also in the way they use the available input
+% information. The syntax is the following:
+%\begin{ttsyntax}
+%\cs{CircleWithCenter}\meta{center} Radius\meta{Radius}
+%\cs{Circlewithcenter}\meta{center} radius\meta{radius}
+%\end{ttsyntax}
+% where in both cases \meta{center} is a vector/ordered couple
+% that points to the circle center. On the contrary \meta{Radius}
+% is a vector obtained through previous calculations, while
+% \meta{radius} is a scalar containing a previously calculated length.
+%    \begin{macrocode}
+\def\CircleWithCenter#1Radius#2{\put(#1){\ModOfVect#2to\CWR
+\circle{\fpeval{2*\CWR}}}\ignorespaces}
+% 
+\def\Circlewithcenter#1radius#2{\put(#1){\circle{\fpeval{2*abs(#2)}}}%
+\ignorespaces}
+%    \end{macrocode}
+%
+% As announced, here we have a macro to compute the axis of a segment;
+% given two points $P_1$ and $P_2$, for example the end points of a
+% segment, or better the end point of the vector that goes from
+% $P_1$ to $P_2$, the macro determines the segment middle point and
+% a second point the lays on the perpendicular at a distance equal to half
+% the first two points distance; this second point lays at the left of
+% vector $P_2-P_1$, therefore it is important to select the right initial
+% vector, in order to have the second axis point on the desired side.
+%\begin{ttsyntax}
+%\cs{AxisOf}\meta{P1} and\meta{P2} to\meta{Axis1}\meta{Axis2}
+%\end{ttsyntax}
+% Macros |\SegmentCenter| and |\MiddlePointOf| are alias to one another;
+% their syntax is:
+%\begin{ttsyntax}
+%\cs{SegmentCenter}\parg{P1}\parg{P2}to\meta{center}
+%\cs{MiddlePointOf}\parg{P1}\parg{P2}to\meta{center}
+%\end{ttsyntax}
+% \meta{P1}, \meta{p2} and \meta{center} are all vectors.
+%    \begin{macrocode}
+
+\def\AxisOf#1and#2to#3#4{%
+\SubVect#1from#2to\Base \ScaleVect\Base by0.5to\Base
+\AddVect\Base and#1to#3 \MultVect\Base by0,1to#4}
+
+\def\SegmentCenter(#1)(#2)to#3{\AddVect#1and#2to\Segm
+\ScaleVect\Segm by0.5to#3\ignorespaces}
+
+\let\MiddlePointOf\SegmentCenter
+%    \end{macrocode}
+%
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \subsection{Triangle special points}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Here we have the macros to find the special points on a triangle
+% side that are the “foot” of special lines from one vertex to the
+% opposite side. We already described the circumcircle and the
+% circumcenter, but that is a separate case, because the circumcenter
+% is not the intersection of special lines from one vertex to the
+% opposite base. The special lines we are interested in here are
+% the height, the median, and the bisector 
+% The macros have the same aspect |\Triangle...Base|, where the dots
+% are replaced with each of the (capitalised) special line names.
+% Their syntaxes are therefore very similar:
+%\begin{ttsyntax}
+%\cs{TriangleMedianBase}\meta{vertex} on\meta{base1} and\meta{base2} to\meta{M}
+%\cs{TriangleHeightBase}\meta{vertex} on\meta{base1} and\meta{base2} to\meta{H}
+%\cs{TrinagleBisectorBase}\meta{vertex} on\meta{base1} and\meta{base2} to\meta{B}
+%\end{ttsyntax}
+% where \meta{vertex} contains one of the vertices coordinates, and
+% \meta{base1} and \meta{base2} are the end points of the side
+% opposite to that triangle vertex; \meta{M}, meta{H}, and \meta{B}
+% are the intersections of these special lines from the \meta{vertex}
+% to the opposite side; in order, they are the foot of the median,
+% the foot of the height; the foot of the bisector. The construction
+% of the median foot \meta{M} is trivial because this foot is the base
+% center; the construction of the height foot is a little more
+% complicated, because it is necessary to find the exact direction
+% of the perpendicular from the vertex to the base in order to
+% find the intersection \meta{H}; the construction of the bisector
+% base implies finding the exact direction of the two sides starting at the
+% \meta{vertex}, and taking the mean direction, which is trivial if
+% polar coordinates are used; at this point the bisector line is completely
+% determined and the intersection with  the base line \meta{B} is
+% easily obtained.
+%    \begin{macrocode}
+
+\def\TriangleMedianBase#1on#2and#3to#4{%
+\SubVect#1from#2to\TMBu \SubVect#1from#3to\TMBd
+\SubVect\TMBu from\TMBd to\Base
+\ScaleVect\Base by0.5to\TMBm\AddVect#2and\TMBm to#4\ignorespaces}
+%
+\def\TriangleHeightBase#1on#2and#3to#4{%
+\SubVect#2from#3to\Base
+\ArgOfVect\Base to\Ang \CopyVect\fpeval{\Ang+90}:1 to\Perp
+\IntersectionOfLines(#1)(\Perp)and(#2)(\Base)to#4\ignorespaces}
+%
+\def\TriangleBisectorBase#1on#2and#3to#4{%
+\SubVect#2from#1to\Luno \SubVect#3from#1to\Ldue
+\SubVect#2from#3to\Base
+\ArgOfVect\Luno to\Arguno \ArgOfVect\Ldue to\Argdue
+\edef\ArgBis{\fpeval{(\Arguno+\Argdue)/2}}%
+\CopyVect \ArgBis:1to \Bisect
+\IntersectionOfLines(#2)(\Base)and(#1)(\Bisect)to#4\ignorespaces}
+%    \end{macrocode}
+% Having defined the previous macros, it becomes very easy to create
+% the macros to find the\emph{barycenter}, the \emph{orthocenter},
+% the\emph{incenter}; for the \emph{circumcenter} and the
+% \emph{circumcircle} we have already solved the question with the
+% |\ThreePointCircleCenter| and the |ThreePointCircle| macros; for
+% homogeneity, we create here their aliases with the same form as
+% the new “center”  macros. Actually, for the “circle” macros,
+% once the center is known, there is no problem with the circumcircle,
+% while for the incircle it suffices a macro to determine the distance
+% of the incenter from one of the triangle sides; such a macro is going to
+% be defined in a little while; it is more general than simply to
+% determine the radius of the incircle.
+%    \begin{macrocode}
+
+\let\TriangleCircumcenter\ThreePointCircleCenter
+\let\TriangleCircummcircle\ThreePointCircle
+%    \end{macrocode}
+%
+% The other “center” macros are the following; they all consist
+% in finding two of the specific triangle lines, and finding their
+% intersection. Therefore for the barycenter we intersect two
+% median lines; for the orthocenter we intersect two height lines;
+% for the incenter we intersect two bisector lines;
+%    \begin{macrocode}
+
+\def\TriangleBarycenter(#1)(#2)(#3)to#4{%
+\TriangleMedianBase#1on#2and#3to\Pa 
+\TriangleMedianBase#2on#3and#1to\Pb
+\DistanceAndDirOfVect#1minus\Pa to\ModPa and\AngPa
+\DistanceAndDirOfVect#2minus\Pb to\ModPb and\AngPb
+\IntersectionOfLines(#1)(\AngPa)and(#2)(\AngPb)to#4}
+
+\def\TriangleOrthocenter(#1)(#2)(#3)to#4{%
+\TriangleHeightBase#1on#2and#3to\Pa 
+\TriangleHeightBase#2on#3and#1to\Pb 
+\DistanceAndDirOfVect#1minus\Pa to\ModPa and\AngPa
+\DistanceAndDirOfVect#2minus\Pb to\ModPb and\AngPb
+\IntersectionOfLines(#1)(\AngPa)and(#2)(\AngPb)to#4}
+
+\def\TriangleIncenter(#1)(#2)(#3)to#4{%
+\TriangleBisectorBase#1on#2and#3to\Pa
+\TriangleBisectorBase#2on#3and#1to\Pb
+\DistanceAndDirOfVect#1minus\Pa to\ModPa and\AngPa
+\DistanceAndDirOfVect#2minus\Pb to\ModPb and\AngPb
+\IntersectionOfLines(#1)(\AngPa)and(#2)(\AngPb)to#4}
+%    \end{macrocode}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%\subsection{Other specific service macros}\label{ssec:ellissi}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% And here it comes the general macro to determine the distance
+% of a point from a segment or from a line that contains that
+% segment; it may be used for determining the radius of the
+% incenter, but it is going to be used also for other purposes.
+% Its syntax is the following:
+%\begin{ttsyntax}
+%\cs{DistanceOfPoint}\meta{point} from\parg{P1}\parg{P2}to\meta{distance}
+%\end{ttsyntax}
+% where \meta{point} is a generic point; \meta{P1} and \meta{P2} 
+% are a segment end points, or two generic points on a line;
+% \meta{distance} is the macro that receives the computed scalar
+% distance value.
+%    \begin{macrocode}
+
+\def\DistanceOfPoint#1from(#2)(#3)to#4{%
+\SubVect#2from#3to\Base \MultVect\Base by0,1to\AB
+\IntersectionOfLines(#1)(\AB)and(#2)(\Base)to\D
+\SubVect#1from\D to\D
+\ModOfVect\D to#4}
+%    \end{macrocode}
+% The following macros are specific to solve other little geometrical
+% problems that arise when creating more complicated constructions.
+%
+% The |\AxisFromAxisAndFocus| is an unhappy name that describes
+% the solution of an ellipse relationship between the ellipse axes
+% and the focal distance 
+%\begin{equation} a^2 = b^2 + c^2\label{equ:axes-foci}\end{equation}
+% This relation exists between the “semi” values, but it works equally
+% well with the full values. Evidently $a$ is the largest quantity and
+% refers to the main ellipse axis, the one that passes through the two
+% foci; $b$ refers to the other shorter ellipse axis and $c$ refers to
+% the foci; $b$ and $c$ are smaller than $a$, but there is no specific
+% relationship among these two quantities It goes by itself that
+% these statements apply to a veritable ellipse, not to a circle,
+% that is the special case where $b=a$ and $c=0$.
+%
+% Since to solve the above equation we have one unknown and two
+% known data, but we do not know what they represent, we have to
+% assume some relationship exist between the known data; therefore
+% if $a$ is known it must be entered as the first macro argument;
+% otherwise $a$ is the unknown and the first Argument has to be the
+% smaller one among $b$ and $c$. Since $b$ and$c$ may come from other
+% computation the user has a dilemma: which is the smaller one?
+% But this is a wrong approach; of course if the user knows which is the
+% smaller, s/he can use the macro by entering the data in the proper
+% order; but the user is determining the main axis, therefore it better
+% that s/he uses directly the second macro |\MainAxisFromAxisAndFocus|
+% that directly computes $a$ disregarding the order with which $b$
+% and $c$ are entered; the macro name suggests to enter $b$ first
+% and $c$ second, but it is irrelevant thanks to the sum properties.
+% Summarising:
+%\begin{itemize}[noitemsep]
+%\item if the main axis is known use |\AxisFromAxisAndFocus| by
+% entering the main axis as the first argument; otherwise
+%\item ~
+%  \begin{itemize}
+%    \item if it is known which is smaller among $b$ and $c$, it is 
+%      possible to use |\AxisFromAxisAndFocus| by entering the smaller
+%      one as the first argument; otherwise
+%    \item determine the main axis by using |\MainAxisFromAxisAndFocus|
+%  \end{itemize}
+%\end{itemize}
+% Their syntaxes of these two commands are basically the following:
+%\begin{ttsyntax}
+%\cs{AxisFromAxisAndFocus}\meta{main axis} and\meta{axis or focus} to\meta{focus or axis}
+%\cs{MainAxisFromAxisAndFocus}\meta{axis or focus} and\meta{focus or axis} to\meta{main axis}
+%\end{ttsyntax}
+% but it is possible to enter the data in a different way with the
+% first command; the described syntax is the suggested one.
+% Evidently \meta{axis or focus} and \meta{focus or axis} imply
+% that if you specify the focus in one of the two, you have to
+% specify the axis in the other one.
+%    \begin{macrocode}
+
+\def\AxisFromAxisAndFocus#1and#2to#3{%
+\fptest{abs(#1)>abs(#2)}%
+   {\edef#3{\fpeval{sqrt(#1**2-#2**2)}}}%
+   {\edef#3{\fpeval{sqrt(#2**2+#1**2)}}}}
+
+\def\MainAxisFromAxisAndFocus#1and#2to#3{%
+\edef#3{\fpeval{sqrt(#2**2+#1**2)}}}
+%    \end{macrocode}
+% The following macros allow to determine some scalar values relative
+% to segments; in the second one the order of the segment end points is
+% important, because the computed argument refers to the vector $P_2 - P_1$.
+% Their syntaxes are the following:
+%\begin{ttsyntax}
+%\cs{SegmentLength}\parg{P1}\parg{P2}to\meta{length}
+%\cs{SegmentArg}\parg{P1}\parg{P2}to\meta{argument}
+%\end{ttsyntax}
+% Both \meta{length} and \meta{argument} are macros that contain
+% scalar quantities; the argument is in the range
+% $-180^\circ <\Phi \leq +180^\circ$.
+%    \begin{macrocode}
+
+\def\SegmentLength(#1)(#2)to#3{\SubVect#1from#2to\Segm
+\ModOfVect\Segm to#3}
+
+\def\SegmentArg(#1)(#2)to#3{\SubVect#1from#2to\Segm
+\GetCoord(\Segm)\SegmX\SegmY\edef#3{\fpeval{atand(\SegmY,\SegmX)}}%
+\ignorespaces}
+%    \end{macrocode}
+%
+% In the following sections we need some transformations, in
+% particular the affine shear one. The macros we define here are
+% not for general use, but are specific for the purpose of this package.
+% 
+% The fist macro shears a segment, or better a vector that goes
+% from point $P_1$ to point $P_2$ with a horizontal shear
+% factor/angle $\alpha$; the origin of the vector does not vary
+% and remains $P_1$ but the arrow tip of the vector is moved
+% according to the shear factor; in practice this shearing macro
+% is valid only for vectors that start from any point laying on
+% the $x$ axis. The shear factor $\alpha$ is the angle of the
+% \emph{clock wise} rotation vector operator by which the vertical
+% coordinate lines get rotated with respect to their original position.
+% The syntax is the following:
+%\begin{ttsyntax}
+%\cs{ShearVect}\parg{P1}\parg{P2}by\meta{shear} to\meta{vector}
+%\end{ttsyntax}
+% where \meta{P1} and \meta{P2} are the initial and final points of the
+% vector to be sheared with the \meta{shear} angle, and the result is put
+% in the output \meta{vector}
+%    \begin{macrocode}
+
+\def\ShearVect(#1)(#2)by#3to#4{%
+\SubVect#1from#2to\AUX 
+\GetCoord(\AUX)\Aux\Auy
+\edef\Aux{\fpeval{\Aux + #3*\Auy}}%
+\edef\Auy{\fpeval{\Auy}}%
+\AddVect\Aux,\Auy and#1to#4\ignorespaces}
+
+%    \end{macrocode}
+% Again we have another different |\ScaleVector| macro that takes in input
+% the starting and ending points of a vector, and scales the vector
+% independently of the initial point.
+%    \begin{macrocode}
+
+\def\ScaleVector(#1)(#2)by#3to#4{%
+% Scala per il fattore #3 il vettore da #1 a #2
+\SubVect#1from#2to\AUX
+\ScaleVect\AUX by#3to\AUX
+\AddVect\AUX and#1to#4\ignorespaces}
+%    \end{macrocode}
+%
+% The following macro to draw a possibly sheared ellipse appears
+% complicated; but in reality it is not much different from a
+% “normal” ellipse drawing command. In oder to do the whole work
+% the ellipse center is set in the origin of the axes, therefore
+% it is not altered by the shearing process; everything else is
+% horizontally sheared by the shear angle $\alpha$. In particular the
+% 12~nodes and control point that are required by the Bézier
+% splines that draw the four ellipse quarters. It is this multitude
+% of shearing commands that makes the macro mach longer and apparently
+% complicated.
+% The syntax is the following:
+%\begin{ttsyntax}
+%\cs{Sellisse}\meta{$\star$}\marg{h-axis}\marg{v-axis}\oarg{shear}
+% \end{ttsyntax}
+% where the optional asterisk is used to mark and label the Bézier
+% spline nodes and the control points of the possibly sheared ellipse;
+% without the asterisk the ellipse is drawn without any “decoration”;
+% the optional \meta{shear} is as usual the angle of the sheared
+% vertical coordinate lines; its default value is zero.
+%    \begin{macrocode}
+%
+\NewDocumentCommand\Sellisse{s m m O{0}}{\bgroup
+\CopyVect#2,#3to\Ptr \ScaleVect\Ptr by-1to\Pbl
+\CopyVect#2,-#3to\Pbr \ScaleVect\Pbr by-1to\Ptl
+\edef\Ys{\fpeval{tand{#4}}}%
+\edef\K{\fpeval{4*(sqrt(2)-1)/3}}%
+%
+\ShearVect(0,0)(0,#3)by\Ys to\Pmt
+\ShearVect(0,0)(0,-#3)by\Ys to\Pmb
+\ShearVect(0,0)(#2,0)by\Ys to\Pmr 
+\ShearVect(0,0)(-#2,0)by\Ys to\Pml
+%
+\ShearVect(\Pmr)(\Ptr)by\Ys to\Ptr 
+\ShearVect(\Pml)(\Ptl)by\Ys to\Ptl 
+\ShearVect(\Pmr)(\Pbr)by\Ys to\Pbr 
+\ShearVect(\Pml)(\Pbl)by\Ys to\Pbl
+%
+\IfBooleanTF{#1}{\Pbox(\Ptr)[bl]{P_{tr}}\Pbox(\Pbl)[tr]{P_{bl}}%
+\Pbox(\Pbr)[tl]{P_{br}}\Pbox(\Ptl)[br]{P_{tl}}%
+\polygon(\Pbr)(\Ptr)(\Ptl)(\Pbl)}{}% 
+%
+\ScaleVector(\Pmr)(\Ptr)by\K to\Crt 
+\ScaleVector(\Pmr)(\Pbr)by\K to\Crb 
+\ScaleVector(\Pml)(\Ptl)by\K to\Clt 
+\ScaleVector(\Pml)(\Pbl)by\K to\Clb 
+\ScaleVector(\Pmt)(\Ptr)by\K to\Ctr 
+\ScaleVector(\Pmt)(\Ptl)by\K to\Ctl 
+\ScaleVector(\Pmb)(\Pbr)by\K to\Cbr 
+\ScaleVector(\Pmb)(\Pbl)by\K to\Cbl
+% 
+\IfBooleanTF{#1}{%
+ \Pbox(\Crt)[l]{C_{rt}}\Pbox(\Crb)[l]{C_{rb}}
+ \Pbox(\Clt)[r]{C_{lt}}\Pbox(\Clb)[r]{C_{lb}}
+ \Pbox(\Ctr)[b]{C_{tr}}\Pbox(\Ctl)[b]{C_{tl}}
+ \Pbox(\Cbr)[t]{C_{br}}\Pbox(\Cbl)[t]{C_{bl}}
+%
+\Pbox(\Pmr)[l]{P_{mr}}\Pbox(\Pmt)[b]{P_{mt}}%
+\Pbox(\Pml)[r]{P_{ml}}\Pbox(\Pmb)[t]{P_{mb}}%
+%
+\polygon(\Pbr)(\Ptr)(\Ptl)(\Pbl)\thicklines}{}%
+%
+\moveto(\Pmr)
+\curveto(\Crt)(\Ctr)(\Pmt)
+\curveto(\Ctl)(\Clt)(\Pml)
+\curveto(\Clb)(\Cbl)(\Pmb)
+\curveto(\Cbr)(\Crb)(\Pmr)
+\fillstroke
+\egroup} 
+
+%    \end{macrocode}
+% This user macro is used to call the |\Sellisse| macro with
+% the desired parameters, but also to act with it on order
+% to fill or stroke the ellipse contour, and to select some
+% settings such as the contour line thickness, or the color
+% of the ellipse contour or interior.
+% the syntax is the following:
+%\begin{ttsyntax}
+%\cs{XSellisse}\meta{$\star$1}\parg{center}\oarg{angle}\aarg{shear}\marg{h-axis}\marg{v axis}\meta{$\star$2}\oarg{settings1}\oarg{settings2}
+%\end{ttsyntax}
+% where there are two optional asterisks, \meta{$\star$1} and
+% \meta{$\star$2}; the first one controls the coloring of the ellipse:
+% if present the interior is filled, if absent the contour is stroked; the
+% second one controls the way a possibly sheared ellipse appears:
+% if present, the construction is shown, if absent only the final result
+% is shown; \meta{center} is optional: if present, the ellipse center is
+% specified; if absent, its center is at the origin of the picture axes;
+% \meta{angle} is optional with default value zero: if absent,
+% the ellipse is not rotated and the \meta{h-axis} remains horizontal,
+% while the \meta{v-axis} remains vertical, while if present and with a
+% non vanishing value, the ellipse is rotated counterclockwise
+% the amount specified, and, of course, if the value is negative,
+% the rotation is clockwise. The optional parameter \meta{shear},
+% if present, shears the ellipse paralle the \meta{h-axis} direction;
+% the \meta{settings1} and \meta{settings2} operate as described
+% for command \cs{Xellisse}.
+%    \begin{macrocode}
+
+\NewDocumentCommand\XSellisse{ s D(){0,0} O{0} D<>{0} m m s O{} o }%
+    {\IfBooleanTF#1{\let\fillstroke\fillpath}%
+                   {\let\fillstroke\strokepath}%
+    \put(#2){\rotatebox{#3}{#8\relax
+    \IfBooleanTF{#7}{\Sellisse*{#5}{#6}[#4]}%
+                  {\Sellisse{#5}{#6}[#4]}%
+    \IfValueTF{#9}{\let\fillstroke\strokepath
+                   #9\Sellisse{#5}{#7}[#4]}{}}}%
+    \ignorespaces}
+%    \end{macrocode}
+%
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \subsection{Regular polygons and special ellipses}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% We finally arrive to more complex macros used to create special
+% polygons and special ellipses.
+%
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \subsubsection{Regular polygons}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Regular polygons are not that special; it is possible to draw them
+% by using the |\multiput| or |\xmultiput| commands, but a single
+% command that does everything by itself with more built in
+% functionalities is much handier. The new command |\RegPolygon|
+% has the following syntax:
+%\begin{ttsyntax}
+%\cs{RegPoligon}\meta{$\star$}\parg{center}\marg{radius}\marg{number}\oarg{angle}\aarg{settings}
+%\end{ttsyntax}
+% where \meta{$\star$} is an optional asterisk; its presence means
+% that the  polygon interior is filled, instead of the polygon
+% contour being stroked; the \meta{center} specification of the
+% polygon is optional; if it is omitted, the polygon center goes
+% to the origin of the |picture| coordinates; \meta{radius} is the
+% mandatory radius of the circumscribed circle, or, in other words,
+% the distance of each polygon vertex form the \meta{center}; the
+% mandatory \meta{number} is an integer that specifies the number of
+% polygon sides; the first vertex that is being drawn by this
+% command, has an angle of zero degrees with respect to the
+% \meta{center}; if a different initial \meta{angle} different from
+% zero is desired, it is specified through this optional argument;
+% possibly the angle bracketed optional \meta{setting} parameter
+% may be used to specify, for example, the line thickness for the
+% contour, and/or the color for the polygon contour or interior.
+% See the documentation \file{euclideangeometry-man.pdf} for more
+% information and usage examples. 
+%    \begin{macrocode}
+
+\NewDocumentCommand\RegPolygon{s D(){0,0}  m m O{0} D<>{\relax} }{{%
+\countdef\I=258 \I=0
+\CopyVect#5:#3to\P
+\CopyVect\fpeval{360/#4}:1to\R
+\put(#2){#6\relax
+         \moveto(\P)\fpdowhile{\I < #4}%
+         {\MultVect\P by\R to\P%\GetCoord(\P)\X\Y
+         \lineto(\P)\advance\I by 1}%
+         \IfBooleanTF{#1}%
+         {\fillpath}{#6\strokepath}}}\ignorespaces}
+%%%%%%%%%
+\def\DirOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
+\ModOfVect#1to\@tempa
+\unless\ifdim\@tempa\p@=\z@
+   \DividE\t@X by\@tempa to\t@X
+   \DividE\t@Y by\@tempa to\t@Y
+\fi\MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
+%    \end{macrocode}
+%
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \subsubsection{The Steiner ellipse}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% The construction of the Steiner ellipse is very peculiar; it is
+% almost intuitive that any triangle has infinitely many internal
+% tangent ellipses; therefore it is necessary to state some other
+% constraints to find one specific ellipse out from this unlimited set.
+%
+% One such ellipse is the Steiner one, obtained by adding the
+% constraint that the ellipse be tangent to the median points 
+% of the triangle sides. But one thing is the definition, and
+% another totally different one is to find the parameters of
+% such an ellipse; and working with ruler and compass, it is
+% necessary to find a procedure to draw such an ellipse.
+%
+% The construction described here and implemented with the
+% |SteinerEllipse| macro is based on the following steps, each
+% one requiring the use of some of the commands and/or
+% transformations described in the previous sections.
+%\begin{enumerate}[noitemsep]
+%^^A
+%\item Given a generic triangle (the coordinates of its three
+% vertices) it is not necessary, but it is clearer to explain,
+% if the triangle is shifted and rotated so as to have one of
+% its sides horizontal, and the third vertex in the upper part
+% of the |picture| drawing. So we first perform the initial
+% shift and rotation and memorise the parameters of this
+% transformation so that, at the end of the procedure, we can
+% put back the triangle (and its Steiner ellipse) in its 
+% original position. Let us call this shifted and rotated triangle
+% with the symbol $T_0$.
+%^^A
+%\item We transform  $T_0$ with an affine shear transformation into an
+% isosceles triangle $T_1$ that has the same base and the same
+% height as $T_0$. We memorise the shear  “angle” so as to proceed
+% to an inverse transformation when the following steps are completed:
+% let be $\alpha$ this shear angle; geometrically it represents the
+% angle of the sheared vertical coordinate lines with respect
+% to the original vertical position.\label{enum:shear}
+%^^A
+%\item With another affine vertical scaling transformation we transform $T_1$ into an equilateral triangle $T_2$; the ratio of the vertical
+% transformation equals the ratio between the $T_2$  to the
+% $T_1$ heights; we memorise this ratio for the reverse transformation
+% at the end of the procedure.\label{enum:ratio}
+%^^A
+%\item The Steiner ellipse of the equilateral triangle $T_2$ is its
+% incircle. We are almost done; we just have to proceed to the
+% inverse transformations; getting back from $T_2$ to $T_1$ first implies
+% transforming the incircle of $T_2$ into an ellipse with its
+% vertical axis scaled by the inverse ratio memorised in
+% step~\ref{enum:ratio}.
+%^^A
+%\item The second inverse transformation by the shear angle is easy
+% with the passage from $T_1$ to $T_0$, but it would be more difficult
+% for transforming the ellipse into the sheared ellipse. We have already
+% defined the |\Sellipse| and the |\XSellipse| macros that may take
+% care of the ellipse  shear transformation; we already memorised the
+% shear angle in step~\ref{enum:shear}, therefore the whole procedure,
+% except for putting back the triangle, is almost done.
+%^^A
+%\item Eventually we perform the last shifting and rotating transformation
+% and the whole construction is completed.
+%^^A
+%\end{enumerate}
+%
+% The new macro Steiner ellipse has therefore the following syntax:
+%\begin{ttsyntax}
+%\cs{SteinerEllipse}\meta{$\star$}\parg{P1}\parg{P2}\parg{P3}\oarg{diameter}
+%\end{ttsyntax}
+% where \meta{P1}, \meta{P2}, \meta{P3} are the vertices of the
+% triangle; \meta{$\star$}  is an optional asterisk; without it
+% the maro draws only the final result, that contains only the
+% given triangle and its Steiner ellipse; on the opposite, if the asterisk
+% is used the whole construction from $T_0$ to its Steiner ellipse
+% is drawn; the labelling of points is done with little dots of
+% the default \meta{diameter} or a specified value; by default it is a 1\,pt
+% diameter, but sometimes it would be better to use a slightly
+% larger value (remembering that 1\,mm  — about three points — 
+% is already too much). 
+% Please refer to the documentation file \file{euclideangeometry-man.pdf}
+% for usage examples and suggestions.
+%    \begin{macrocode}
+%
+
+\NewDocumentCommand\SteinerEllipse{s d() d() d() O{1}}{\bgroup
+%
+\IfBooleanTF{#1}{}{\put(#2)}{%
+  \CopyVect0,0to\Pu  
+  \SubVect#2from#3to\Pd 
+  \SubVect#2from#4to\Pt 
+  \ModAndAngleOfVect\Pd to\M and\Rot 
+  \MultVect\Pd by-\Rot:1 to\Pd  \MultVect\Pt by-\Rot:1 to\Pt
+  \IfBooleanTF{#1}{}{\rotatebox{\Rot}}{\makebox(0,0)[bl]{%
+  \Pbox(\Pu)[r]{P_1}[#5]<-\Rot>\Pbox(\Pd)[t]{P_2}[#5]<-\Rot>
+  \Pbox(\Pt)[b]{P_3}[#5]<-\Rot>%
+  \polygon(\Pu)(\Pd)(\Pt)%
+  \edef\B{\fpeval{\M/2}}\edef\H{\fpeval{\B*tand(60)}}
+  \IfBooleanTF{#1}{\Pbox(\B,\H)[b]{H}[#5]
+  \polygon(\Pu)(\B,\H)(\Pd)}{}%
+  \edef\R{\fpeval{\B*tand(30)}}
+  \IfBooleanTF{#1}{\Pbox(\B,\R)[bl]{C}[#5]
+    \Circlewithcenter\B,\R radius{\R}}{}%
+  \GetCoord(\Pt)\Xt\Yt\edef\VScale{\fpeval{\Yt/\H}}
+  \IfBooleanTF{#1}{\polyline(\Pu)(\B,\Yt)(\Pd)
+  \Pbox(\B,\Yt)[b]{V}[#5]}{}%
+  \edef\Ce{\fpeval{\R*\VScale}}
+  \IfBooleanTF{#1}{\Xellisse(\B,\Ce){\R}{\Ce}
+    \Pbox(\B,\Ce)[r]{C_e}[#5]\Pbox(\B,0)[t]{B}[#5]}{}%
+  \SubVect\B,0 from\Pt to\SlMedian 
+  \IfBooleanTF{#1}{\Dotline(\B,0)(\Pt){2}[1.5]}{}%
+  \ModAndAngleOfVect\SlMedian to\Med and\Alfa 
+  \edef\Alfa{\fpeval{90-\Alfa}} 
+    \IfBooleanTF{#1}{\Dotline(\B,\Yt)(\B,0){2}[1.5]
+    \Pbox(\fpeval{\B+\Ce*tand{\Alfa}},\Ce)[l]{C_i}[#5]
+    \VectorArc(\B,0)(\B,15){-\Alfa}
+    \Pbox(\fpeval{\B+2.5},14)[t]{\alpha}[0]}{}%
+  \edef\a{\R}\edef\b{\Ce}%
+\CopyVect\fpeval{\B+\Ce*tand{\Alfa}},\Ce to\CI 
+\XSellisse(\CI)<\Alfa>{\R}{\Ce}
+}}}%
+\egroup\ignorespaces}
+\let\EllisseSteiner\SteinerEllipse
+%    \end{macrocode}
+%
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%\subsubsection{The ellipse that is internally tangent to a triangle while one of its foci is prescribed}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% We now are going to tackle another problem. As we said before, any
+% triangle has an infinite set of internally tangent circles, unless
+% some further constraint is specified. 
+%
+% Another problem of this kind is the determination and geometrical
+% construction of an internally tangent ellipse when one focus is
+% specified; of course since the whole ellipse is totally internal
+% to the triangle, we assume that the user has already verified 
+% that the coordinates of the focus fall inside the triangle.
+% We are not going to check this feature in place of the user;
+% after all, if the user draws the triangle within a |picture| image,
+% together with the chosen focus, is suffices a glance to verify that
+% such focus lays within the triangle perimeter.
+%
+% The geometrical construction is quite complicated, but it is
+% described in a paper by Estevão V.~Candia on \TB~2019 \textbf{40}(3);
+% it consists of the following steps.
+%\begin{enumerate}[noitemsep]
+%^^A
+%\item Suppose you have specified a triangle by means of its three
+% vertices, and a point inside it to play the role of a focus; it
+% is necessary to find the other focus and the main axis length in
+% order to have a full description of the ellipse.
+%^^A
+%\item To do so, it is necessary to find the focus three symmetrical
+% points with respect to the three sides.
+%^^A
+%\item The center of the three point circle through these symmetrical
+% points is the second focus.
+%^^A
+%\item  The lines that join the second focus to the three symmetrical
+% points of the first focus, intersect the triangle sides in three
+% points that result to be the tangency points of the ellipse to the
+% triangle.
+%^^A
+%\item Chosen one of these tangency points and computing the sum of its
+% distances from both foci, the total length of the ellipsis  main axis
+% is found.
+%^^A
+%\item Knowing both foci, the total inter focal distance is found,
+% therefore equation~\eqref{equ:axes-foci} allows to find the other
+% axis length.
+%^^A
+%\item The inclination of the focal segment gives us the the rotation
+% to which the ellipse is subject, and the middle point of such
+% segment gives the ellipse center.
+%^^A
+%\item At this point we have all the necessary elements to draw
+% the ellipse.
+%^^A
+%\end{enumerate}
+%
+% We need another little macro to find the symmetrical points;
+% if the focus $F$ and its symmetrical point $P$ with respect
+% to a side/segment, the intersection of such segment $F-P$ with
+% the side is the segment middle point $M$; from this property
+% we derive the formula $P= 2M -F$. Now $M$  is also the
+% intersection of the line passing through $F$ and perpendicular
+% to the side. Therefore it is particularly simple to compute,
+% but its better to have available a macro that does the whole
+% work; here it is, but it assumes the the center of symmetry is
+% already known:
+%    \begin{macrocode}
+
+\def\SymmetricalPointOf#1respect#2to#3{\ScaleVect#2by2to\Segm
+\SubVect#1from\Segm to#3\ignorespaces}
+%    \end{macrocode}
+% And its syntax is the following:
+%\begin{ttsyntax}
+%\cs{SymmetricalPointOf}\meta{focus} respect\meta{symmetry center}
+%\qquad to\meta{symmetrical point}
+%\end{ttsyntax}
+% where the argument names are self explanatory.
+%
+% The overall macro that executes all the passages described in
+% the above enumeration follows; the reader can easily recognise
+% the various steps, since the names of the macros are self
+% explanatory; the $G_i$ point names are the symmetrical ones
+% to the first focus $F$; the $M_i$ points are the centers of
+% symmetry; the $F'$ point is the second focus; the $T_i$ points
+% are the tangency points. The macro |\EllipseWithFOcus| has the
+% following syntax: 
+%\begin{ttsyntax}
+%\cs{EllipseWithFocus}\meta{$\star$}\parg{P1}\parg{P2}\parg{P3}\parg{focus}
+%\end{ttsyntax}
+% where \meta{P1}, \meta{P2}, \meta{P3}  are the triangle vertices
+% and  \meta{focus} contains the first focus coordinates; the
+% optional asterisk, as usual, selects the construction steps
+% versus the final result: no asterisk, no construction steps.
+%    \begin{macrocode}
+
+\NewDocumentCommand\EllipseWithFocus{s d() d() d() d()}{\bgroup%
+\CopyVect#2to\Pu 
+\CopyVect#3to\Pd 
+\CopyVect#4to\Pt 
+\CopyVect#5to\F 
+\polygon(\Pu)(\Pd)(\Pt)
+\Pbox(\Pu)[r]{P_1}[1.5pt]\Pbox(\Pd)[t]{P_2}[1.5pt]
+\Pbox(\Pt)[b]{P_3}[1.5pt]\Pbox(\F)[b]{F}[1.5pt]
+\SegmentArg(\Pu)(\Pt)to\At
+\SegmentArg(\Pu)(\Pd)to\Ad
+\SegmentArg(\Pd)(\Pt)to\Au
+\IntersectionOfLines(\Pu)(\At:1)and(\F)(\fpeval{\At+90}:1)to\Mt
+\IntersectionOfLines(\Pd)(\Ad:1)and(\F)(\fpeval{\Ad+90}:1)to\Md
+\IntersectionOfLines(\Pd)(\Au:1)and(\F)(\fpeval{\Au+90}:1)to\Mu
+\IfBooleanTF{#1}{\Pbox(\Mt)[br]{M_3}[1.5pt]\Pbox(\Md)[t]{M_2}[1.5pt]
+  \Pbox(\Mu)[b]{M_1}[1.5pt]}{}
+\SymmetricalPointOf\F respect\Mu to\Gu 
+\IfBooleanTF{#1}{\Pbox(\Gu)[l]{G_1}[1.5pt]}{}
+\SymmetricalPointOf\F respect \Md to\Gd 
+\IfBooleanTF{#1}{\Pbox(\Gd)[t]{G_2}[1.5pt]}{}
+\SymmetricalPointOf\F respect \Mt to\Gt 
+\IfBooleanTF{#1}{\Pbox(\Gt)[r]{G_3}[1.5pt]}{}
+\IfBooleanTF{#1}{\ThreePointCircle*(\Gu)(\Gd)(\Gt)}%
+                {\ThreePointCircle(\Gu)(\Gd)(\Gt)}
+\CopyVect\C to\Fp \Pbox(\Fp)[l]{F'}[1.5pt]
+\IfBooleanTF{#1}{%
+\Dotline(\F)(\Gt){2}[1.5pt] 
+\Dotline(\F)(\Gd){2}[1.5pt]
+\Dotline(\F)(\Gu){2}[1.5pt]}{}
+\IntersectionOfSegments(\Pu)(\Pt)and(\Fp)(\Gt)to\Tt
+\IntersectionOfSegments(\Pu)(\Pd)and(\Fp)(\Gd)to\Td
+\IntersectionOfSegments(\Pd)(\Pt)and(\Fp)(\Gu)to\Tu
+\IfBooleanTF{#1}{\Pbox(\Tu)[l]{T_1}[1.5pt]
+\Pbox(\Td)[b]{T_2}[1.5pt]
+\Pbox(\Tt)[tl]{T_3}[1.5pt]
+\Dashline(\Fp)(\Gu){1}\Dashline(\Fp)(\Gd){1}\Dashline(\Fp)(\Gt){1}}{}
+\DistanceAndDirOfVect\Fp minus\Tt to\DFp and\AFu 
+\DistanceAndDirOfVect\F  minus\Tt to\DF and\AF 
+\SegmentCenter(\F)(\Fp)to\CE \Pbox(\CE)[b]{C}[1.5pt]
+\edef\a{\fpeval{(\DFp+\DF)/2}} 
+\SegmentArg(\F)(\Fp)to\AngFocalAxis  
+\SegmentLength(\F)(\CE)to\c
+\AxisFromAxisAndFocus\a and\c to\b 
+\Xellisse(\CE)[\AngFocalAxis]{\a}{\b}[\thicklines]
+\VECTOR(-30,0)(120,0)\Pbox(120,0)[t]{x}[0]
+\VECTOR(0,-20)(0,130)\Pbox(0,130)[r]{y}[0]\Pbox(0,0)[tr]{O}[1.5pt]
+\egroup\ignorespaces}
+\let\EllisseConFuoco\EllipseWithFocus
+%    \end{macrocode}
+%
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \section{Comments on this package}
+%^^A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% In general we found very comfortable to draw ellipses and
+% to define macros to draw not only such shapes or filled elliptical
+% areas, but also to create “legends” with coloured backgrounds and
+% borders; such applications found their way in other works.
+% But here we dealt with other geometrical problems. The accompanying
+% document |euclideangeometry-man.pdf| describes much clearly with
+% examples  what you can do with the macros described in this package.
+% In facts, this file just describes the package macros, and it gives
+% some ideas on how to extend the ability of |curve2e| to draw geometrical
+% diagrams.
+% The users who would like to modify or to add some functionalities
+% are invited to do so; I will certainly acknowledge their contributions
+% and even add their names to the list of authors.
+%
+% As long as I can, I enjoy playing with \LaTeX and its wonderful
+% facilities; but, taking into consideration my age, I would invite
+% the users to consider the possibility of assuming the maintenance
+% of this package.
+%
+%\iffalse
+%</package>
+%\fi
+%
+%\iffalse
+%<*readme>
+%^^A\fi
+% \section{The \texttt{README.txt} file}
+% The following is the text that forms the contents of the |README.txt|
+% file that accompanies the package. We found it handy to have it in
+% the documented source, because in this way certain pieces of
+% information don't need to be repeated again and again in different
+% files. 
+%    \begin{macrocode}
+The package bundle euclideangeometry is composed of the following files
+
+euclideangeometry.dtx
+euclideangeometry.pdf
+euclideangeometry-man.tex
+euclideangeometry-man.pdf
+README.txt
+
+euclideangeometry.dtx is the documented TeX source file of package 
+euclideangeometry.sty; you get euclideangeometry.sty,
+euclideangeometry.pdf by running pdflatex on euclideangeometry.dtx. 
+
+README.txt, this file, contains general information.
+
+euclideangeometry-man.tex and euclideangeometry-man.pdf are 
+the source file and the readable document containing the end 
+user manual. 
+In other words euclideangeometry.pdf is oriented towards the 
+developers and euclideangeometry-man.pdf to the end users.
+
+
+Claudio Beccari
+
+claudio dot beccari at gmail dot com
+%    \end{macrocode}
+%^^A\iffalse
+%</readme>
+%\fi
+% \Finale
+%^^A <driver>\endinput
diff --git a/Master/texmf-dist/tex/latex/euclideangeometry/euclideangeometry.sty b/Master/texmf-dist/tex/latex/euclideangeometry/euclideangeometry.sty
new file mode 100644
index 00000000000..77807562370
--- /dev/null
+++ b/Master/texmf-dist/tex/latex/euclideangeometry/euclideangeometry.sty
@@ -0,0 +1,518 @@
+%%
+%% This is file `euclideangeometry.sty',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% euclideangeometry.dtx  (with options: `package')
+%% 
+%%   Copyright (C)  2020 Claudio Beccari all rights reserved.
+%%   License information appended
+%% 
+\NeedsTeXFormat{LaTeX2e}[2019/01/01]
+\ProvidesPackage{euclideangeometry}%
+        [2020-02-02 v.0.1.2 Extension package for curve2e]
+
+\makeatletter
+\providecommand\GetFileInfo{}
+\renewcommand\GetFileInfo[1]{%
+  \def\filename{#1}%
+  \def\@tempb##1 v.##2 ##3\relax##4\relax{%
+    \def\filedate{##1}\def\fileversion{##2}\def\fileinfo{##3}}%
+  \edef\@tempa{\csname ver@#1\endcsname}%
+  \expandafter\@tempb\@tempa\relax? ? \relax\relax}
+\def\cleanfiledate#1-#2-#3!{\unless\ifx#3\empty
+\edef\FileDate{#1#2#3}\else\unless\ifx#2\empty
+\edef\FileDate{#1#2}\else\edef\FileDate{#1}\fi\fi}
+\RequirePackage{curve2e}
+\GetFileInfo{curve2e.sty}
+\expandafter\cleanfiledate\filedate!
+
+\ifnum\FileDate<20200118
+\typeout{*************************************************************}
+\typeout{Package curve2e too old}
+\typeout{Be sure that your TeX installation is complete and up to date}
+\typeout{*************************************************************}
+\typeout{Input of euclideangeometry stopped and job aborted}
+\typeout{*************************************************************}
+\expandafter\@@end\fi
+
+\providecommand\file{}
+\renewcommand*\file[1]{{\normalfont\texttt{#1}}}
+\providecommand\prog{}
+\renewcommand*\prog[1]{{\normalfont\texttt{#1}}}
+\newcommand*\conta[1]{\texttt{\slshape#1}}
+\providecommand\cs{}
+\renewcommand\cs[1]{{\normalfont\texttt{\char92#1}}}
+\providecommand\meta{}
+\renewcommand*\meta[1]{{\normalfont$\langle$\textit{#1}$\rangle$}}
+\providecommand\marg{}
+\renewcommand*\marg[1]{{\ttfamily \char123\meta{#1}\char125}}
+\newcommand*\Marg[1]{{\normalfont\ttfamily\{#1\}}}\let\Arg\Marg
+\providecommand\oarg{}
+\renewcommand*\oarg[1]{{\ttfamily[\meta{#1}]}}
+\newcommand*\Oarg[1]{{\ttfamily[#1]}}
+\newcommand*\aarg[1]{{\ttfamily{<\meta{#1}>}}}
+\newcommand*\Aarg[1]{{\ttfamily<#1>}}
+\newcommand*\Bambiente[1]{\texttt{\char92begin\{#1\}}}
+\newcommand*\Eambiente[1]{\texttt{\char92end\{#1\}}}
+\newcommand*\garg[1]{\texttt{(\splitgarg#1,!)}}\let\parg\garg
+\newcommand*\Garg[1]{\texttt{(#1)}}\let\Parg\Garg
+  \def\secgarg#1,{#1}
+  \def\splitgarg#1,#2!{%
+     \ifstrequal{#2}{}{\meta{#1}}{\meta{#1},\meta{\secgarg#2}}}
+\providecommand\pack{}
+\renewcommand*\pack[1]{{\normalfont\textsf{#1}}}
+\providecommand\opt{}
+\renewcommand*\opt[1]{{\normalfont\textsl{#1}}}\let\opz\opt
+\providecommand\env{}
+\renewcommand*\env[1]{{\normalfont\textit{\bfseries#1}}}\let\amb\env
+\providecommand*\ped[1]{\ensuremath{_{\mathrm{#1}}}}
+\providecommand*\ap[1]{\ensuremath{^{\mathrm{#1}}}}
+\providecommand*\unit[1]{\ensuremath{\,\mathrm{#1}}}
+\providecommand*\sigla[1]{\textsc{\lowercase{#1}}}
+\providecommand*\iu{\ensuremath{\mathrm{i}}}\let\gei\iu
+\providecommand*\eu{\ensuremath{\mathop{\mathrm{e}}\nolimits}}
+\DeclareRobustCommand*{\Ars}{%
+  \textsf{\lower -.48ex\hbox{\rotatebox{-20}{A}}\kern -.3em{rs}}%
+  \kern -.05em\TeX\unskip\kern -.17em\lower -.357ex\hbox{nica}\xspace}
+
+\DeclareRobustCommand*\LyX{%
+  L\kern-0.2em\raisebox{-0.5ex}{Y}\kern-0.2em{X}\xspace}
+
+\definecolor{verdeguit}{rgb}{0, 0.40, 0}
+\RequirePackage{iftex}
+\def\GuIT{\mbox{\color{verdeguit}\def\I{\textcolor{black}{I}}%
+\ifPDFTeX
+  \usefont{T1}{lmr}{m}{sc}
+\else
+  \usefont{TU}{lmr}{m}{sc}
+\fi
+g\raisebox{-0.715ex}{\kern-0.26em u}\kern-0.13em\I\kern-0.14em t}\xspace}
+\providecommand*\usecs[1]{\csname#1\endcsname}
+\providecommand*\whilenum[2]{\usecs{@whilenum}#1\do{#2}}
+\RequirePackage{afterpage}
+\providecommand\supaginapari[1]{%
+  \afterpage{%
+    \ifodd\value{page}\expandafter\@firstoftwo
+    \else\expandafter\@secondoftwo
+    \fi{\supaginapari{#1}}{#1}
+  }
+}\let\onevenpage\supaginapari
+\providecommand\supaginadispari[1]{%
+  \afterpage{%
+    \ifodd\value{page}\expandafter\@secondoftwo
+    \else\expandafter\@firstoftwo
+    \fi{\supaginadispari{#1}}{#1}
+  }
+}\let\onoddpage\supaginadispari
+\RequirePackage{etoolbox}
+\AfterEndPreamble{%
+\apptocmd{\TeX}{{}\xspace}{}{}
+\apptocmd{\LaTeX}{{}\xspace}{}{}}
+
+\DeclareRobustCommand*\TeXLive{\TeX\ Live\xspace}
+\DeclareRobustCommand*\MiKTeX{MiK\TeX}
+\DeclareRobustCommand*\MacTeX{Mac\TeX}
+\DeclareRobustCommand*\pdfLaTeX{pdf\/\LaTeX}
+\DeclareRobustCommand*\LuaLaTeX{Lua\LaTeX}
+\DeclareRobustCommand*\XeLaTeX{X\ifdim\fontdimen1\font=0pt\kern-0.15em\fi
+\lower.5ex\hbox{\rotatebox[origin=c]{180}{E}}%
+\ifdim\fontdimen1\font=0pt\kern-0.15em\else\kern-0.275em\fi
+\LaTeX}
+\DeclareRobustCommand*\TikZ{Ti\emph{k}Z\xspace}
+\DeclareRobustCommand*\TUGboat{\textsl{TUG\-boat}\xspace}
+\renewcommand\dbltopfraction{1.}
+\def\hz{\hskip0pt}
+\newenvironment{ttsintassi}{\begin{lrbox}{0}
+\minipage{\dimexpr\linewidth-2\fboxrule-2\fboxsep}\ttfamily\obeylines}%
+{\endminipage\end{lrbox}\center\fbox{\box0}\endcenter}
+\let\ttsyntax\ttsintassi \let\endttsyntax\endttsintassi
+
+\unless\ifcsname Gr@phGrid\endcsname
+\let\originalGraphGrid\GraphGrid
+\RenewDocumentCommand\GraphGrid{r() d()}{%
+\IfValueTF{#2}{\put(#1){\originalGraphGrid(#2)}}%
+              {\put(0,0){\originalGraphGrid(#1)}}}\fi
+\providecommand\Pbox{}
+\RenewDocumentCommand\Pbox{D(){0,0} O{cc} m O{0.5ex} s D<>{0}}{%
+\put(#1){\rotatebox{#6}{\makebox(0,0){%
+\dimendef\Dim=2566\relax
+\settowidth\Dim{#2}%
+\edef\Rapp{\fpeval{\Dim/{1ex}}}%
+\fptest{\Rapp > 1.5}{\fboxsep=0.5ex}{\fboxsep=0.75ex}%
+\IfBooleanTF{#5}{\fboxrule=0.4pt}{\fboxrule=0pt}%
+\fptest{#4 = 0sp}%
+  {\makebox(0,0)[#2]{\fbox{$\relax#3\relax$}}}%
+  {\edef\Diam{\fpeval{(#4)/\unitlength}}%
+   \makebox(0,0){\circle*{\Diam}}%
+   \makebox(0,0)[#2]{\fbox{$\relax\mathsf#3\relax$}}%
+  }}}%
+}\ignorespaces}
+\DeclareRobustCommand\setfontsize[2][1.2]{%
+  \linespread{#1}\fontsize{#2}{#2}\selectfont}
+
+\def\EUGsplitArgs(#1,#2)#3#4{\edef#3{#1}\edef#4{#2}}
+
+\providecommand\Zbox{}
+\RenewDocumentCommand\Zbox{R(){0,0} D(){0,0} O{bl} m}{%
+\EUGsplitArgs(#2)\ZboxX\ZboxY % separa la x e la y della scatola
+\fboxsep=2\unitlength
+\ifnum\ZboxX=\csuse{z@}
+  \def\ZTesto{\fbox{#4}}%
+\else
+  \ifnum\ZboxY=\csuse{z@}
+    \def\ZTesto{\fbox{\parbox{\ZboxX\unitlength}{#4}}}%
+  \else
+    \def\ZTesto{%
+    \setbox2560=\hbox{\fbox{%
+         \parbox[c][\ZboxY\unitlength][c]{\ZboxX\unitlength}{#4}}}%
+      \dimen2560=\dimexpr(\ht2560 +\dp2560)/2\relax
+      \ht2560=\dimen2560\relax
+      \dp2560=\dimen2560\relax
+      \box2560%
+      }%
+  \fi
+\fi
+\put(#1){\makebox(0,0)[#3]{\ZTesto}}\ignorespaces}
+
+\RequirePackage{verbatim}
+\newwrite\example@out
+\ProvideDocumentEnvironment{Esempio}{s O{\normalsize} D(){0.40}}
+{\par\addvspace{3.0ex plus 0.8ex minus 0.5ex}\vskip -\parskip
+\dimendef\Wboxu=2570 \dimendef\Wboxd=2572
+\Wboxu=#3\textwidth\relax
+\Wboxd=\dimexpr\linewidth-\columnsep-\Wboxu\relax
+\begingroup
+\@bsphack
+\immediate\openout\example@out\jobname-temp.tex
+\let\do\@makeother\dospecials\catcode`\^^M\active
+\def\verbatim@processline{%
+  \immediate\write\example@out{\the\verbatim@line}}%
+\verbatim@start\relax}%
+{\immediate\closeout\example@out\@esphack\endgroup
+\begin{minipage}{\textwidth}%
+\IfBooleanTF{#1}{\begin{minipage}{\textwidth}}{\begin{minipage}{\Wboxu}}%
+#2\relax
+\verbatiminput{\jobname-temp.tex}
+\end{minipage}%
+\IfBooleanTF{#1}{\par\bigskip}{\hfill}%
+\IfBooleanTF{#1}{\begin{minipage}{\textwidth}}{\begin{minipage}{\Wboxd}}%
+\raggedleft
+\input{\jobname-temp}
+\end{minipage}
+\end{minipage}\par
+}
+
+
+\NewDocumentCommand\ellisse{ s m m}{%
+\IfBooleanTF{#1}%
+  {\let\fillstroke\fillpath}%
+  {\let\fillstroke\strokepath}%
+\Sellisse{#2}{#3}%
+}
+
+\NewDocumentCommand\Xellisse{ s D(){0,0} O{0} m m O{} o}{%
+\IfBooleanTF{#1}%
+  {\XSellisse*(#2)[#3]{#4}{#5}[#6][#7]}%
+  {\XSellisse(#2)[#3]{#4}{#5}[#6][#7]}%
+}
+
+\def\polyvector(#1){\roundcap\def\EUGpreviouspoint{#1}\EUGpolyvector}
+\def\EUGpolyvector(#1){%
+\@ifnextchar({%
+    \segment(\EUGpreviouspoint)(#1)\def\EUGpreviouspoint{#1}\EUGpolyvector}%
+  {\VECTOR(\EUGpreviouspoint)(#1)}%
+}
+
+
+\def\IntersectionOfLines(#1)(#2)and(#3)(#4)to#5{\bgroup
+\def\IntPu{#1}\def\Uu{#2}\def\IntPd{#3}\def\Ud{#4}%
+ \DirOfVect\Uu to\Du
+ \DirOfVect\Ud to\Dd
+ \XpartOfVect\Du to \a  \YpartOfVect\Du to \b
+ \XpartOfVect\Dd to \c  \YpartOfVect\Dd to \d
+ \XpartOfVect\IntPu to \xu \YpartOfVect\IntPu to \yu
+ \XpartOfVect\IntPd to \xd \YpartOfVect\IntPd to \yd
+ \edef\Den{\fpeval{-(\a*\d-\b*\c)}}%
+ \fptest{abs(\Den)<1e-5}{% almost vanishing determinant
+   \def#5{0,0}%
+ }{% Determinant OK
+   \edef\Numx{\fpeval{(\c*(\b*\xu-\a*\yu)-\a*(\d*\xd-\c*\yd))/\Den}}%
+   \edef\Numy{\fpeval{(\d*(\b*\xu-\a*\yu)-\b*(\d*\xd-\c*\yd))/\Den}}%
+   \CopyVect\Numx,\Numy to\Paux
+   \edef\x{\egroup\noexpand\edef\noexpand#5{\Paux}}\x\ignorespaces}}
+
+\def\IntersectionOfSegments(#1)(#2)and(#3)(#4)to#5{%
+\SubVect#1from#2to\IoSvectu \DirOfVect\IoSvectu to\DirIoSVecu
+\SubVect#3from#4to\IoSvectd \DirOfVect\IoSvectd to\DirIoSVecd
+\IntersectionOfLines(#1)(\DirIoSVecu)and(#3)(\DirIoSVecd)to#5\ignorespaces}
+
+\NewDocumentCommand\ThreePointCircle{s d() d() d()}{%
+\AxisOf#2and#3to\Mu\Du \AxisOf#2and#4to\Md\Dd
+\IntersectionOfLines(\Mu)(\Du)and(\Md)(\Dd)to\C
+\SubVect#2from\C to\R
+\IfBooleanTF{#1}{\CircleWithCenter\C Radius\R}{}\ignorespaces}
+
+\NewDocumentCommand\ThreePointCircleCenter{d() d() d() m}{%
+\ThreePointCircle(#1)(#2)(#3)\CopyVect\C to#4}
+\def\CircleWithCenter#1Radius#2{\put(#1){\ModOfVect#2to\CWR
+\circle{\fpeval{2*\CWR}}}\ignorespaces}
+\def\Circlewithcenter#1radius#2{\put(#1){\circle{\fpeval{2*abs(#2)}}}%
+\ignorespaces}
+
+\def\AxisOf#1and#2to#3#4{%
+\SubVect#1from#2to\Base \ScaleVect\Base by0.5to\Base
+\AddVect\Base and#1to#3 \MultVect\Base by0,1to#4}
+
+\def\SegmentCenter(#1)(#2)to#3{\AddVect#1and#2to\Segm
+\ScaleVect\Segm by0.5to#3\ignorespaces}
+
+\let\MiddlePointOf\SegmentCenter
+
+\def\TriangleMedianBase#1on#2and#3to#4{%
+\SubVect#1from#2to\TMBu \SubVect#1from#3to\TMBd
+\SubVect\TMBu from\TMBd to\Base
+\ScaleVect\Base by0.5to\TMBm\AddVect#2and\TMBm to#4\ignorespaces}
+\def\TriangleHeightBase#1on#2and#3to#4{%
+\SubVect#2from#3to\Base
+\ArgOfVect\Base to\Ang \CopyVect\fpeval{\Ang+90}:1 to\Perp
+\IntersectionOfLines(#1)(\Perp)and(#2)(\Base)to#4\ignorespaces}
+\def\TriangleBisectorBase#1on#2and#3to#4{%
+\SubVect#2from#1to\Luno \SubVect#3from#1to\Ldue
+\SubVect#2from#3to\Base
+\ArgOfVect\Luno to\Arguno \ArgOfVect\Ldue to\Argdue
+\edef\ArgBis{\fpeval{(\Arguno+\Argdue)/2}}%
+\CopyVect \ArgBis:1to \Bisect
+\IntersectionOfLines(#2)(\Base)and(#1)(\Bisect)to#4\ignorespaces}
+
+\let\TriangleCircumcenter\ThreePointCircleCenter
+\let\TriangleCircummcircle\ThreePointCircle
+
+\def\TriangleBarycenter(#1)(#2)(#3)to#4{%
+\TriangleMedianBase#1on#2and#3to\Pa
+\TriangleMedianBase#2on#3and#1to\Pb
+\DistanceAndDirOfVect#1minus\Pa to\ModPa and\AngPa
+\DistanceAndDirOfVect#2minus\Pb to\ModPb and\AngPb
+\IntersectionOfLines(#1)(\AngPa)and(#2)(\AngPb)to#4}
+
+\def\TriangleOrthocenter(#1)(#2)(#3)to#4{%
+\TriangleHeightBase#1on#2and#3to\Pa
+\TriangleHeightBase#2on#3and#1to\Pb
+\DistanceAndDirOfVect#1minus\Pa to\ModPa and\AngPa
+\DistanceAndDirOfVect#2minus\Pb to\ModPb and\AngPb
+\IntersectionOfLines(#1)(\AngPa)and(#2)(\AngPb)to#4}
+
+\def\TriangleIncenter(#1)(#2)(#3)to#4{%
+\TriangleBisectorBase#1on#2and#3to\Pa
+\TriangleBisectorBase#2on#3and#1to\Pb
+\DistanceAndDirOfVect#1minus\Pa to\ModPa and\AngPa
+\DistanceAndDirOfVect#2minus\Pb to\ModPb and\AngPb
+\IntersectionOfLines(#1)(\AngPa)and(#2)(\AngPb)to#4}
+
+\def\DistanceOfPoint#1from(#2)(#3)to#4{%
+\SubVect#2from#3to\Base \MultVect\Base by0,1to\AB
+\IntersectionOfLines(#1)(\AB)and(#2)(\Base)to\D
+\SubVect#1from\D to\D
+\ModOfVect\D to#4}
+
+\def\AxisFromAxisAndFocus#1and#2to#3{%
+\fptest{abs(#1)>abs(#2)}%
+   {\edef#3{\fpeval{sqrt(#1**2-#2**2)}}}%
+   {\edef#3{\fpeval{sqrt(#2**2+#1**2)}}}}
+
+\def\MainAxisFromAxisAndFocus#1and#2to#3{%
+\edef#3{\fpeval{sqrt(#2**2+#1**2)}}}
+
+\def\SegmentLength(#1)(#2)to#3{\SubVect#1from#2to\Segm
+\ModOfVect\Segm to#3}
+
+\def\SegmentArg(#1)(#2)to#3{\SubVect#1from#2to\Segm
+\GetCoord(\Segm)\SegmX\SegmY\edef#3{\fpeval{atand(\SegmY,\SegmX)}}%
+\ignorespaces}
+
+\def\ShearVect(#1)(#2)by#3to#4{%
+\SubVect#1from#2to\AUX
+\GetCoord(\AUX)\Aux\Auy
+\edef\Aux{\fpeval{\Aux + #3*\Auy}}%
+\edef\Auy{\fpeval{\Auy}}%
+\AddVect\Aux,\Auy and#1to#4\ignorespaces}
+
+
+\def\ScaleVector(#1)(#2)by#3to#4{%
+\SubVect#1from#2to\AUX
+\ScaleVect\AUX by#3to\AUX
+\AddVect\AUX and#1to#4\ignorespaces}
+\NewDocumentCommand\Sellisse{s m m O{0}}{\bgroup
+\CopyVect#2,#3to\Ptr \ScaleVect\Ptr by-1to\Pbl
+\CopyVect#2,-#3to\Pbr \ScaleVect\Pbr by-1to\Ptl
+\edef\Ys{\fpeval{tand{#4}}}%
+\edef\K{\fpeval{4*(sqrt(2)-1)/3}}%
+\ShearVect(0,0)(0,#3)by\Ys to\Pmt
+\ShearVect(0,0)(0,-#3)by\Ys to\Pmb
+\ShearVect(0,0)(#2,0)by\Ys to\Pmr
+\ShearVect(0,0)(-#2,0)by\Ys to\Pml
+\ShearVect(\Pmr)(\Ptr)by\Ys to\Ptr
+\ShearVect(\Pml)(\Ptl)by\Ys to\Ptl
+\ShearVect(\Pmr)(\Pbr)by\Ys to\Pbr
+\ShearVect(\Pml)(\Pbl)by\Ys to\Pbl
+\IfBooleanTF{#1}{\Pbox(\Ptr)[bl]{P_{tr}}\Pbox(\Pbl)[tr]{P_{bl}}%
+\Pbox(\Pbr)[tl]{P_{br}}\Pbox(\Ptl)[br]{P_{tl}}%
+\polygon(\Pbr)(\Ptr)(\Ptl)(\Pbl)}{}%
+\ScaleVector(\Pmr)(\Ptr)by\K to\Crt
+\ScaleVector(\Pmr)(\Pbr)by\K to\Crb
+\ScaleVector(\Pml)(\Ptl)by\K to\Clt
+\ScaleVector(\Pml)(\Pbl)by\K to\Clb
+\ScaleVector(\Pmt)(\Ptr)by\K to\Ctr
+\ScaleVector(\Pmt)(\Ptl)by\K to\Ctl
+\ScaleVector(\Pmb)(\Pbr)by\K to\Cbr
+\ScaleVector(\Pmb)(\Pbl)by\K to\Cbl
+\IfBooleanTF{#1}{%
+ \Pbox(\Crt)[l]{C_{rt}}\Pbox(\Crb)[l]{C_{rb}}
+ \Pbox(\Clt)[r]{C_{lt}}\Pbox(\Clb)[r]{C_{lb}}
+ \Pbox(\Ctr)[b]{C_{tr}}\Pbox(\Ctl)[b]{C_{tl}}
+ \Pbox(\Cbr)[t]{C_{br}}\Pbox(\Cbl)[t]{C_{bl}}
+\Pbox(\Pmr)[l]{P_{mr}}\Pbox(\Pmt)[b]{P_{mt}}%
+\Pbox(\Pml)[r]{P_{ml}}\Pbox(\Pmb)[t]{P_{mb}}%
+\polygon(\Pbr)(\Ptr)(\Ptl)(\Pbl)\thicklines}{}%
+\moveto(\Pmr)
+\curveto(\Crt)(\Ctr)(\Pmt)
+\curveto(\Ctl)(\Clt)(\Pml)
+\curveto(\Clb)(\Cbl)(\Pmb)
+\curveto(\Cbr)(\Crb)(\Pmr)
+\fillstroke
+\egroup}
+
+
+\NewDocumentCommand\XSellisse{ s D(){0,0} O{0} D<>{0} m m s O{} o }%
+    {\IfBooleanTF#1{\let\fillstroke\fillpath}%
+                   {\let\fillstroke\strokepath}%
+    \put(#2){\rotatebox{#3}{#8\relax
+    \IfBooleanTF{#7}{\Sellisse*{#5}{#6}[#4]}%
+                  {\Sellisse{#5}{#6}[#4]}%
+    \IfValueTF{#9}{\let\fillstroke\strokepath
+                   #9\Sellisse{#5}{#7}[#4]}{}}}%
+    \ignorespaces}
+
+\NewDocumentCommand\RegPolygon{s D(){0,0}  m m O{0} D<>{\relax} }{{%
+\countdef\I=258 \I=0
+\CopyVect#5:#3to\P
+\CopyVect\fpeval{360/#4}:1to\R
+\put(#2){#6\relax
+         \moveto(\P)\fpdowhile{\I < #4}%
+         {\MultVect\P by\R to\P%\GetCoord(\P)\X\Y
+         \lineto(\P)\advance\I by 1}%
+         \IfBooleanTF{#1}%
+         {\fillpath}{#6\strokepath}}}\ignorespaces}
+%%%%%%%%%
+\def\DirOfVect#1to#2{\GetCoord(#1)\t@X\t@Y
+\ModOfVect#1to\@tempa
+\unless\ifdim\@tempa\p@=\z@
+   \DividE\t@X by\@tempa to\t@X
+   \DividE\t@Y by\@tempa to\t@Y
+\fi\MakeVectorFrom\t@X\t@Y to#2\ignorespaces}%
+
+\NewDocumentCommand\SteinerEllipse{s d() d() d() O{1}}{\bgroup
+\IfBooleanTF{#1}{}{\put(#2)}{%
+  \CopyVect0,0to\Pu
+  \SubVect#2from#3to\Pd
+  \SubVect#2from#4to\Pt
+  \ModAndAngleOfVect\Pd to\M and\Rot
+  \MultVect\Pd by-\Rot:1 to\Pd  \MultVect\Pt by-\Rot:1 to\Pt
+  \IfBooleanTF{#1}{}{\rotatebox{\Rot}}{\makebox(0,0)[bl]{%
+  \Pbox(\Pu)[r]{P_1}[#5]<-\Rot>\Pbox(\Pd)[t]{P_2}[#5]<-\Rot>
+  \Pbox(\Pt)[b]{P_3}[#5]<-\Rot>%
+  \polygon(\Pu)(\Pd)(\Pt)%
+  \edef\B{\fpeval{\M/2}}\edef\H{\fpeval{\B*tand(60)}}
+  \IfBooleanTF{#1}{\Pbox(\B,\H)[b]{H}[#5]
+  \polygon(\Pu)(\B,\H)(\Pd)}{}%
+  \edef\R{\fpeval{\B*tand(30)}}
+  \IfBooleanTF{#1}{\Pbox(\B,\R)[bl]{C}[#5]
+    \Circlewithcenter\B,\R radius{\R}}{}%
+  \GetCoord(\Pt)\Xt\Yt\edef\VScale{\fpeval{\Yt/\H}}
+  \IfBooleanTF{#1}{\polyline(\Pu)(\B,\Yt)(\Pd)
+  \Pbox(\B,\Yt)[b]{V}[#5]}{}%
+  \edef\Ce{\fpeval{\R*\VScale}}
+  \IfBooleanTF{#1}{\Xellisse(\B,\Ce){\R}{\Ce}
+    \Pbox(\B,\Ce)[r]{C_e}[#5]\Pbox(\B,0)[t]{B}[#5]}{}%
+  \SubVect\B,0 from\Pt to\SlMedian
+  \IfBooleanTF{#1}{\Dotline(\B,0)(\Pt){2}[1.5]}{}%
+  \ModAndAngleOfVect\SlMedian to\Med and\Alfa
+  \edef\Alfa{\fpeval{90-\Alfa}}
+    \IfBooleanTF{#1}{\Dotline(\B,\Yt)(\B,0){2}[1.5]
+    \Pbox(\fpeval{\B+\Ce*tand{\Alfa}},\Ce)[l]{C_i}[#5]
+    \VectorArc(\B,0)(\B,15){-\Alfa}
+    \Pbox(\fpeval{\B+2.5},14)[t]{\alpha}[0]}{}%
+  \edef\a{\R}\edef\b{\Ce}%
+\CopyVect\fpeval{\B+\Ce*tand{\Alfa}},\Ce to\CI
+\XSellisse(\CI)<\Alfa>{\R}{\Ce}
+}}}%
+\egroup\ignorespaces}
+\let\EllisseSteiner\SteinerEllipse
+
+\def\SymmetricalPointOf#1respect#2to#3{\ScaleVect#2by2to\Segm
+\SubVect#1from\Segm to#3\ignorespaces}
+
+\NewDocumentCommand\EllipseWithFocus{s d() d() d() d()}{\bgroup%
+\CopyVect#2to\Pu
+\CopyVect#3to\Pd
+\CopyVect#4to\Pt
+\CopyVect#5to\F
+\polygon(\Pu)(\Pd)(\Pt)
+\Pbox(\Pu)[r]{P_1}[1.5pt]\Pbox(\Pd)[t]{P_2}[1.5pt]
+\Pbox(\Pt)[b]{P_3}[1.5pt]\Pbox(\F)[b]{F}[1.5pt]
+\SegmentArg(\Pu)(\Pt)to\At
+\SegmentArg(\Pu)(\Pd)to\Ad
+\SegmentArg(\Pd)(\Pt)to\Au
+\IntersectionOfLines(\Pu)(\At:1)and(\F)(\fpeval{\At+90}:1)to\Mt
+\IntersectionOfLines(\Pd)(\Ad:1)and(\F)(\fpeval{\Ad+90}:1)to\Md
+\IntersectionOfLines(\Pd)(\Au:1)and(\F)(\fpeval{\Au+90}:1)to\Mu
+\IfBooleanTF{#1}{\Pbox(\Mt)[br]{M_3}[1.5pt]\Pbox(\Md)[t]{M_2}[1.5pt]
+  \Pbox(\Mu)[b]{M_1}[1.5pt]}{}
+\SymmetricalPointOf\F respect\Mu to\Gu
+\IfBooleanTF{#1}{\Pbox(\Gu)[l]{G_1}[1.5pt]}{}
+\SymmetricalPointOf\F respect \Md to\Gd
+\IfBooleanTF{#1}{\Pbox(\Gd)[t]{G_2}[1.5pt]}{}
+\SymmetricalPointOf\F respect \Mt to\Gt
+\IfBooleanTF{#1}{\Pbox(\Gt)[r]{G_3}[1.5pt]}{}
+\IfBooleanTF{#1}{\ThreePointCircle*(\Gu)(\Gd)(\Gt)}%
+                {\ThreePointCircle(\Gu)(\Gd)(\Gt)}
+\CopyVect\C to\Fp \Pbox(\Fp)[l]{F'}[1.5pt]
+\IfBooleanTF{#1}{%
+\Dotline(\F)(\Gt){2}[1.5pt]
+\Dotline(\F)(\Gd){2}[1.5pt]
+\Dotline(\F)(\Gu){2}[1.5pt]}{}
+\IntersectionOfSegments(\Pu)(\Pt)and(\Fp)(\Gt)to\Tt
+\IntersectionOfSegments(\Pu)(\Pd)and(\Fp)(\Gd)to\Td
+\IntersectionOfSegments(\Pd)(\Pt)and(\Fp)(\Gu)to\Tu
+\IfBooleanTF{#1}{\Pbox(\Tu)[l]{T_1}[1.5pt]
+\Pbox(\Td)[b]{T_2}[1.5pt]
+\Pbox(\Tt)[tl]{T_3}[1.5pt]
+\Dashline(\Fp)(\Gu){1}\Dashline(\Fp)(\Gd){1}\Dashline(\Fp)(\Gt){1}}{}
+\DistanceAndDirOfVect\Fp minus\Tt to\DFp and\AFu
+\DistanceAndDirOfVect\F  minus\Tt to\DF and\AF
+\SegmentCenter(\F)(\Fp)to\CE \Pbox(\CE)[b]{C}[1.5pt]
+\edef\a{\fpeval{(\DFp+\DF)/2}}
+\SegmentArg(\F)(\Fp)to\AngFocalAxis
+\SegmentLength(\F)(\CE)to\c
+\AxisFromAxisAndFocus\a and\c to\b
+\Xellisse(\CE)[\AngFocalAxis]{\a}{\b}[\thicklines]
+\VECTOR(-30,0)(120,0)\Pbox(120,0)[t]{x}[0]
+\VECTOR(0,-20)(0,130)\Pbox(0,130)[r]{y}[0]\Pbox(0,0)[tr]{O}[1.5pt]
+\egroup\ignorespaces}
+\let\EllisseConFuoco\EllipseWithFocus
+%% 
+%% Distributable under the LaTeX Project Public License,
+%% version 1.3c or higher (your choice). The latest version of
+%% this license is at: http://www.latex-project.org/lppl.txt
+%% 
+%% This work is "maintained"
+%% 
+%% This work consists of files:
+%% 1) euclideangeometry.dtx, and the derived files euclideangeometry.sty
+%%                           and euclideangeometry.pdf,
+%% 2) euclideangeometry-man.tex and the derived file euclideangeometry-man.pdf,
+%% 3) the auxiliary derived file README.txt
+%% 
+%%
+%% End of file `euclideangeometry.sty'.
diff --git a/Master/tlpkg/bin/tlpkg-ctan-check b/Master/tlpkg/bin/tlpkg-ctan-check
index 9238b256537..27402fea3f3 100755
--- a/Master/tlpkg/bin/tlpkg-ctan-check
+++ b/Master/tlpkg/bin/tlpkg-ctan-check
@@ -272,7 +272,7 @@ my @TLP_working = qw(
     eso-pic esrelation esstix estcpmm esvect
     etaremune etdipa etex-pkg etexcmds etextools ethiop ethiop-t1
     etoc etoolbox etoolbox-de etsvthor
-    euenc euflag eukdate
+    euclideangeometry euenc euflag eukdate
     euler eulerpx eulervm euro euro-ce europasscv europecv eurosym
     everyhook everypage
     exam exam-n exam-randomizechoices examdesign example examplep
diff --git a/Master/tlpkg/tlpsrc/collection-mathscience.tlpsrc b/Master/tlpkg/tlpsrc/collection-mathscience.tlpsrc
index afe27e23816..bbdcceb62a6 100644
--- a/Master/tlpkg/tlpsrc/collection-mathscience.tlpsrc
+++ b/Master/tlpkg/tlpsrc/collection-mathscience.tlpsrc
@@ -78,6 +78,7 @@ depend engtlc
 depend eqexpl
 depend eqnarray
 depend eqnnumwarn
+depend euclideangeometry
 depend extarrows
 depend extpfeil
 depend faktor
diff --git a/Master/tlpkg/tlpsrc/euclideangeometry.tlpsrc b/Master/tlpkg/tlpsrc/euclideangeometry.tlpsrc
new file mode 100644
index 00000000000..e69de29bb2d
--- /dev/null
+++ b/Master/tlpkg/tlpsrc/euclideangeometry.tlpsrc