euclideangeometry (6feb20)

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+% \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