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\documentclass[12pt, draft]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}
\usepackage[a4paper]{geometry}
\usepackage[usenames]{pstcol}
\usepackage{pst-eucl}
\usepackage{pst-plot}
\usepackage{multido}
\usepackage{ifthen}
\usepackage{calc}
\usepackage{array}
\usepackage{moreverb}
\usepackage{multicol}
\usepackage{mathrsfs}
\usepackage[dvips]{changebar}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% QQ DEFINITIONS
\newcommand{\Vecteur}[1]{\ensuremath{\overrightarrow{#1}}}%
\newcommand{\Angle}[1]{\ensuremath{\widehat{#1}}}%
\makeatletter
\newcommand{\Arg}[1]{{\normalfont$\{$\@Arg{#1}$\}$}}%
\newcommand{\Argsans}[1]{\@Arg{#1}}%
\newcommand{\OptArg}[1]{{\normalfont[{\@Arg{#1}}]}}%
\newcommand{\@Arg}[1]%
  {$\protect\langle${\itshape\mdseries\rmfamily#1}$\protect\rangle$}%
\makeatother
%% le backslash \
\newcommand{\bs}{\symbol{'134}}%
\newcommand{\defcom}[2]%
  {\begin{trivlist}\item\fbox{\texttt{\upshape\bs#1}#2}\end{trivlist}}%
\newcommand{\defcomdeux}[4]%
  {%
    \begin{center}%
    \begin{minipage}[t]{.45\linewidth}%
      \begin{trivlist}\item\fbox{\texttt{\upshape\bs#1}#2}\end{trivlist}%
    \end{minipage}%
    \hfill%
    \begin{minipage}[t]{.45\linewidth}%
      \begin{trivlist}\item\fbox{\texttt{\upshape\bs#3}#4}\end{trivlist}%
    \end{minipage}
    \end{center}}%
\newenvironment{tabexemple}[1]%
  [@{}m{.3\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.677\linewidth}@{}]%
  {\vspace{1em}\noindent\small\begin{center}%
   \noindent\begin{tabular}{#1}}%
  {\mbox{}\vspace{-1.5em}\end{tabular}\end{center}}%
\newcommand{\tabex}[2]%
  [@{}m{.3\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.677\linewidth}@{}]%
{%
  \begin{tabexemple}[#1]%
    \input{Exemples/#2}&\verbatiminput{Exemples/#2_in}%
  \end{tabexemple}%
}%
\newcommand{\param}[1]{\texttt{#1}}
\newcommand{\com}[1]{\texttt{\bs #1}}
\newcommand{\DefaultVal}[1]{(\texttt{#1} by default)}
\newcommand{\PStricks}{\texttt{PStricks}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{The \texttt{pst-euclide.sty} Package}
\author{Version $0.\beta.5$\\Dominique Rodriguez}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\geometry{bottom=1cm, left=1cm, right=1cm}
\psset{subgriddiv=0,griddots=5}
%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
\newcounter{i}
%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
\begin{document}
\renewcommand{\abstractname}{WARNING}
\maketitle
\cbstart
\begin{abstract}
  This is a $\beta$-version, it is still in development, the macros
  name can and will change without upward-compatibility.
\end{abstract}
\cbend
\begin{multicols}{2}
\tableofcontents{}
\end{multicols}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Special specifications}

\cbstart
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{\PStricks\ Options}

The package activates the \com{SpecialCoor} mode. This mode extend the
coordinates specification. Furthermore the plotting type is set to
\texttt{dimen=middle}, which indicates that the position of the
drawing is done according to the middle of the line. Please look at
the user manual for more information about these setting.

At last, the working axes are supposed to be (ortho)normed.

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Conventions}

For this manual, I used the geometric French conventions for naming
the points:

\begin{itemize}
\item $O$ is a centre (circle, axes, symmetry, homothety, rotation);
\item $I$ defined the unity of the abscissa axe, or a midpoint;
\item $J$ defined the unity of the ordinate axe;
\item $A$, $B$, $C$, $D$ are points ;
\item $M'$ is the image of $M$ by a transformation ;
\end{itemize}

At last, although these are nodes in \PStricks, I treat them
intentionally as points.
\cbend

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic Objects}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Points}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{default axes}

\defcom{pstGeonode}{\OptArg{par}\Arg{$A$}}

This command defines a geometrical point associated with a node.  This
point has a node name \Argsans{$A$} which defines the default label put on
the picture. This label is managed by default in mathematical mode,
the boolean parameter \param{PtNameMath} can modify this behavior and
let manage the label in normal mode. It is put at a distance of
\texttt{1em} of the center of the node with a angle of
\param{PosAngle}\DefaultVal{0}. It is possible to
specify another label using the parameter \param{PointName}, and an
empty label can be specified by selecting the value \texttt{none}, in
that case the point will have no name on the picture.

The point symbol is given by the parameter
\param{PointSymbol} \DefaultVal{*}.  The symbol is the same as used by
the macro \com{pstdot}.  This parameter can be set to \texttt{none},
which means that neither the point nor its label will be drawn on the
picture.

Here are the possible values for this parameter:

\begin{multicols}{3}
  \begin{itemize}\psset{dotscale=2}
  \item \param{*}: \psdots(.5ex,.5ex)
  \item \param{o}: \psdots[dotstyle=o](.5ex,.5ex)
  \item \param{+}: \psdots[dotstyle=+](.5ex,.5ex)
  \item \param{x}: \psdots[dotstyle=x](.5ex,.5ex)
  \item \param{asterisk} : \psdots[dotstyle=asterisk](.5ex,.5ex)
  \item \param{oplus} : \psdots[dotstyle=oplus](.5ex,.5ex)
  \item \param{otimes} : \psdots[dotstyle=otimes](.5ex,.5ex)
  \item \param{triangle}: \psdots[dotstyle=triangle](.5ex,.5ex)
  \item \param{triangle*}: \psdots[dotstyle=triangle*](.5ex,.5ex)
  \item \param{square}: \psdots[dotstyle=square](.5ex,.5ex)
  \item \param{square*}: \psdots[dotstyle=square*](.5ex,.5ex)
  \item \param{diamond} : \psdots[dotstyle=diamond](.5ex,.5ex)
  \item \param{diamond*} : \psdots[dotstyle=diamond*](.5ex,.5ex)
  \item \param{pentagon}: \psdots[dotstyle=pentagon](.5ex,.5ex)
  \item \param{pentagon*} \psdots[dotstyle=pentagon*](.5ex,.5ex)
  \item \param{|}: \psdots[dotstyle=|](.5ex,.5ex)
  \end{itemize}
\end{multicols}

% EXEMPLE GEONODE
\tabex{geonode}

Obviously, the nodes appearing in the picture can be used as normal
pstricks nodes. Thus, it is possible to reference a point from
\rnode{ici}{here}.
\nccurve{->}{ici}{B_1}

% There is a bug linked to the definition of a point, so a mandatory
% \verb$%$ symbol must be put at the end of each such command. If it is
% forgotten, a shift appear on the picture as it can be seen on the
% following diagram:

% \begin{center}
%   \begin{pspicture}(-2,-2)(2,2)\psgrid
%     \pstGeonode(0,0){A}{A}
%     \pstGeonode[PosAngle=-90](1,2){B1}{B_1}
%     \pstGeonode[PointSymbol=pstSmallCircle,
%       linecolor=red](-2,1){B2}{B_2}%
%   \end{pspicture}
% \end{center}

\cbstart
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{User defined axes}

\defcom{pstOIJGeonode}{\OptArg{par}$(x,y)$\Arg{$A$}\Arg{$O$}\Arg{$I$}\Arg{$J$}}

This command allows the placement of points in any landmark(?) defined
by the three points $(O;I;J)$.

%% EXAMPLE
\tabex{oij}
\cbend
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Segment mark}

A segment can be drawn using the \texttt{\bs ncline} command. However,
for marking a segment there is the following command:

\defcom{pstMarkSegment}{\OptArg{par}\Arg{$A$}\Arg{$B$}}

The symbol drawn on the segment is given by the parameter
\param{SegmentSymbol}. Its value can be any valid command which can be
used in math mode. Its default value is \texttt{pstSlashsSlash},
which produced two slashes on the segment. The segment is drawn.

%% EXAMPLE
\tabex{segmentmark}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Triangles}

The more classical figure, it has its own macro for a quick definition:

\defcom{pstTriangle}{%
  \OptArg{par}
  $(x_A;y_A)$\Arg{$A$}$(x_B;y_B)$\Arg{$B$}$(x_C;y_C)$\Arg{$C$}}

In order to accurately put the name of the points, there are three
parameters \param{PosAngleA}, \param{PosAngleB} and \param{PosAngleC},
which are associated respectively to the nodes \Argsans{$A$},
\Argsans{$B$} et \Argsans{$C$}. Obviously they have the same meaning
as the parameter \param{PosAngle}. If one or more of such parameters is
omitted, the value of \param{PosAngle} is taken.

In the same way there are parameters for controlling the symbol used
for each points: \param{PointSymbolA}, \param{PointSymbolB} and
\param{PointSymbolC}. They are equivalent to the parameter
\param{PointSymbol}. The management of the default value followed the
same rule.

\tabex{triangle}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Angles}
  
Each angle is defined with three points. The vertex is the second
point. Their order is important because it is assumed that the angle is
specified in the direct order. The first command is the marking of a
right angle:

\defcom{pstRightAngle}%
  {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$C$}}

The only parameter controlling this command, excepting the ones which
controlled the line, is \param{RightAngleSize} which defines the size
of the symbol\DefaultVal{0.28 unit}.

For other angles, there is the command:

\defcom{pstMarkAngle}%
  {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$C$}}

The \param{label} can be any valid \TeX\ box, it is put at
\param{LabelSep} \DefaultVal{1 unit} of the node in the direction of
\cbstart the bisector of the angle modified by
\param{LabelAngleOffset}\DefaultVal{0} and positioned using
\param{LabelRefPt} \DefaultVal{c}. \cbend Furthermore the arc used for
marking has a radius of \param{MarkAngleRadius} \DefaultVal{.4~unit}.
At least, it is possible to place an arrow using the parameter
\param{arrows}.

\tabex{angle}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Lines, half-lines and segments}

The classical line!

\defcom{pstLineAB}{\OptArg{par}\Arg{$A$}\Arg{$B$}}

In order to control its length\footnote{which is the comble for a
line!}, the two parameters \param{nodesepA} et \param{nodesepB}
specify the abscissa of the extremity of the drawing part of the line.
A negative abscissa specify an outside point, while a positive
abscissa specify an internal point. If these parameters have to be
equal, \param{nodesep} can be used instead. The default value of these
parameters is equal to 0.

\tabex{droite}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Circles}

A circle can be defined either with its center and a point of its
circumference, or with two diameterly opposed points. There is two
commands :

\defcomdeux{pstCircleOA}{\OptArg{par}\Arg{$O$}\Arg{$A$}}%
           {pstCircleAB}{\OptArg{par}\Arg{$A$}\Arg{$B$}}

For the first macro, it is possible to omit the second point and then
to specify a radius or a diameter using the parameters \param{Radius}
and \param{Diameter}. The values of these parameters must be specified
with one of the two following macros :

\defcomdeux{pstDistAB}{\OptArg{par}\Arg{$A$}\Arg{$B$}}%
           {pstDistVal}{\OptArg{par}\Arg{x}}

The first specifies a distance between two points. The parameter
\param{DistCoef} can be used to specify a coefficient to reduce or
enlarge this distance. To be taken into account this last parameter
must be specified before the distance. The second macro can be used to
specify an explicit numeric value.

We will see later how to draw the circle crossing three points.

\vspace{1.1\baselineskip}
\begin{minipage}[m]{.45\linewidth}
  With this package, it becomes possible to draw:

  \begin{itemize}
  \item {\color{red} the circle of center $A$ crossing $B$ ;}
  \item {\color{green} the circle of center $A$ whose radius is $AC$ ;}
  \item {\color{blue} the circle of center $A$ whose radius is $BC$ ;}
  \item {\color{Sepia} the circle of center $B$ whose radius is $AC$ ;}
  \item {\color{Aquamarine} the circle of center $B$ of diameter $AC$ ;}
  \item {\color{RoyalBlue} the circle whose diameter is $BC$ ;}
  \end{itemize}
\end{minipage}
%
\input{Exemples/cercle}

\verbatiminput{Exemples/cercle_in}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Circle arcs}

\defcomdeux{pstArcOAB}{\OptArg{par}\Arg{$O$}\Arg{$A$}\Arg{$B$}}%
           {pstArcnOAB}{\OptArg{par}\Arg{$O$}\Arg{$A$}\Arg{$B$}}

These two macros draw circle arcs, $O$ is the center, the radius
defined by $OA$, the beginning angle given by $A$ and the final angle
by $B$. Finally, the first macro draws the arc in the direct way,
whereas the second in the indirect way. It is not necessary that the
two points are at the same distance of $O$.

\tabex{arc}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Curved abscissa}

A point can be positioned on a circle using its curved abscissa.

\defcom{pstCurvAbsNode}{\OptArg{par}\Arg{$O$}\Arg{$A$}\Arg{$B$}\Arg{Abs}}

The point \Argsans{$B$} is positioned on the circle of center
\Argsans{$O$} crossing \Argsans{$A$}, with the curved abscissa
\Argsans{Abs}. The origin is \Argsans{$A$} and the direction is
anti-clockwise by default. The parameter \param{CurvAbsNeg}
\DefaultVal{false} can change this behavior.

\tabex{abscur}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Généric curve}

It is possible to generate a set of point using a loop, and to give
them a generic name defined by a radical and a number. The following
command can draw a interpolated curve crossing all such kind of
points.

\defcom{pstGenericCurve}{\OptArg{par}\Arg{Radical}\Arg{$n_1$}\Arg{$n_2$}}

The curve is drawn on the points whose name is defined using the
radical \Argsans{Radical} followed by a number from \Argsans{$n_1$} to
\Argsans{$n_2$}. In order to manage side effect, the parameters
\param{GenCurvFirst} et \param{GenCurvLast} can be used to specified
special first or last point. The parameter \param{GenCurvInc} can be
used to modify the increment from a point to the next one
\DefaultVal{1}.

\tabex{gencur}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Geometric Transformations}

The geometric transformation are the ideal tools to construct
geometric figures. All the classical transformation are accessible
with the following macros.

They share the parameter \param{CodeFig} which draws the specific
constructions lines. Its default value is \param{false}, and a
\param{true} value activates this optional drawing.

The drawing is done using the line style \param{CodeFigStyle}
\DefaultVal{dashed}, with the color \param{CodeFigColor}
\DefaultVal{cyan}.

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Central symmetry}

\defcom{pstSymO}%
  {\OptArg{par}\Arg{$O$}\Arg{$M$}\Arg{$M'$}}

Draw the symmetric point in relation to point $O$. The classical
parameter of point creation are usable here, and also for all the
following functions.

\tabex{symcentrale}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Orthogonal (or axial) symmetry}

\defcom{pstOrtSym}%
  {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$M$}\Arg{$M'$}}

Draw the symmetric point in relation to line $(AB)$.

\tabex{symorthogonale}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Rotation}

\defcom{pstRotation}%
  {\OptArg{par}\Arg{$O$}\Arg{$M$}\Arg{$M'$}}

Draw the image of $M$ by the rotation of center $O$ and angle given by
the parameter \param{RotAngle}. This later can be an angle specified
by three points. In such a case, the following function must be used:

\defcom{pstAngleABC}{\Arg{$A$}\Arg{$B$}\Arg{$C$}}

Never forget to use the rotation for a square or a equilateral
triangle.

\tabex{rotation}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Translation}

\defcom{pstTranslation}%
  {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$M$}\Arg{$M'$}}

Draw the translated of vector \Vecteur{AB} of the point
\Argsans{$M$}. Useful for drawing a parallel line.

\tabex{translation}

The parameter \param{DistCoef} can be used as a multiplicand
coefficient to modify the translation vector.

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Homothetie}

\defcom{pstHomO}%
  {\OptArg{par}\Arg{$O$}\Arg{$M$}\Arg{$M'$}}

Draw $M'$ the image of $M$ by the homotethy of center $O$ and
coefficient specified with the parameter \param{HomCoef}.

\tabex{homothetie}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Orthogonal projection}

\defcom{pstProjection}%
  {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$M$}\Arg{$M'$}}

Project orthogonally the point $M$ on the line 
$(AB)$. Useful for the altitude of a triangle.

\tabex{projection}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Special object}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Midpoint}

\defcom{pstMiddleAB}%
  {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$I$}}

Draw the midpoint $I$ of segment $[AB]$.

\tabex{milieu}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Triangle center of gravity}

\defcom{pstCGravABC}%
  {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$C$}\Arg{$G$}}

Draw the $ABC$ triangle centre of gravity $G$.

\tabex{grav}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Centre of the circumcircle of a triangle}

\defcom{pstCircleABC}{\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$C$}\Arg{$O$}}

It's obvious.

\tabex%
  [@{}m{.35\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.627\linewidth}@{}]%
  {ccirc}

\cbstart
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Perpendicular bisector of a segment}

\defcom{pstMediatorAB}{\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$I$}\Arg{$M$}}

The perpendicular bisector of a segment is a line perpendicular to
this segment in its midpoint. The segment is $[AB]$, the midpoint $I$,
and $M$ is a point belonging to the perpendicular bisector line. It is
build by a rotation of $B$ of 90 degrees around $I$. This mean
that the order of $A$ and $B$ is important, it controls the position
of $M$. The command creates the two points $M$ end $I$. The
construction is controlled by the following parameters:

\begin{itemize}
\item \param{CodeFig}, \param{CodeFigColor} et \param{SegmentSymbol}
  for marking the right angle ;
\item \param{PointSymbol} et \param{PointName} for controlling the
  drawing of the two points, each of them can be specified
  separately with the parameters \param{...A} et \param{...B} ;
\item parameters controlling the line drawing.
\end{itemize}

\tabex%
  [@{}m{.35\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.627\linewidth}@{}]%
  {mediator}
\cbend

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Bissectors of angles}

\defcom{pstBissectBAC}{\OptArg{par}\Arg{$B$}\Arg{$A$}\Arg{$C$}\Arg{$N$}}

\defcom{pstOutBissectBAC}{\OptArg{par}\Arg{$B$}\Arg{$A$}\Arg{$C$}\Arg{$N$}}

there are two bisectors for a given geometric angle: the inside one and
the outside one; this is why there is two commands. The angle is
specified by three points specified in the trigonometric direction
(anti-clockwise). The result of the commands is the specific line and
a point belonging to this line. This point is built by a rotation of
point $B$.

\tabex%
  [@{}m{.35\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.627\linewidth}@{}]%
  {bissec}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Intersections}

Points can be defined by intersections. Three intersection types  are
managed:

\begin{itemize}
\item line-line;
\item line-circle;
\item circle-circle.
\end{itemize}

An intersection can not exist: case of parallel lines. In such a case,
the point(s) are positioned at the origin. In fact, the user has to
manage the existence of these points.

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Line-Line}

\defcom{pstInterLL}%
  {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$C$}\Arg{$D$}\Arg{$M$}}

Draw the intersection point between lines $(AB)$ and $(CD)$.

\begin{description}
\item[basique]

  \tabex{interDD}

\item[Horthocentre]

  \tabex%
  [@{}m{.35\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.627\linewidth}@{}]
    {orthocentre}

\end{description}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Circle--Line}

\defcom{pstInterLC}%
  {\OptArg{par}\Arg{$A$}\Arg{$B$}\Arg{$O$}\Arg{$C$}%
    \Arg{$M_1$}\Arg{$M_2$}}

Draw the one or two intersection point(s) between the line  $(AB)$ and
the circle of centre $O$ and with radius $OC$.

The circle is specified with its center and either a point of its
circumference or with a radius specified with parameter \param{radius}
or its diameter specified with parameter \param{Diameter}. These two
parameters can be modify by coefficient \param{DistCoef}.

\tabex
  [@{}m{.4\linewidth}@{\hspace{.013\linewidth}}>{\small}m{.5777\linewidth}@{}]
  {interDC}

In the case of a two points intersection, their position is not
specified, the user will have to check the result. It can arise that
after a modification the name of points can be swapped leading to a
wrong figure as shown in the following example:

\hfill\rule[-\baselineskip]{0pt}{5cm+2\baselineskip}
\begin{pspicture}(5,5)\psgrid
  \pstGeonode(2,2){O}\pstGeonode(3,1){A}%
  \pstGeonode(1,2){B}\pstGeonode(1.1,4){C}%
  \pstCircleOA{O}{A}%
  \pstInterLC{B}{C}{O}{A}{D}{E}
\end{pspicture}
\hfill
\begin{pspicture}(5,5)\psgrid
  \pstGeonode(2,2){O}\pstGeonode(3,1){A}%
  \pstGeonode(1,2){B}\pstGeonode(.9,4){C}%
  \pstCircleOA{O}{A}%
  \pstInterLC{B}{C}{O}{A}{D}{E}
\end{pspicture}
\hspace*{\fill}

This management is the same for the circle-circle intersection.

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Circle--Circle}

\defcom{pstInterCC}%
  {\OptArg{par}\Arg{$O_1$}\Arg{$B$}\Arg{$O_2$}\Arg{$C$}%
    \Arg{$M_1$}\Arg{$M_2$}}

This function is similar to the last one. The boolean parameters
\param{CodeFigA} et \param{CodeFigB} allow the drawing of the arcs
at the intersection. In order to get a coherence \param{CodeFig} allow
the drawing of both arcs. The boolean parameters \param{CodeFigAarc} and
\param{CodeFigBarc} specified the direction of these optional arcs:
trigonometric (by default) or clockwise. Here is a first example.

\tabex{interCC}

And a more complete one, which includes the special circle
specification using radius and diameter. For such specifications it
exists the parameters \param{RadiusA}, \param{RadiusB},
\param{DiameterA} and \param{DiameterB}.

\begin{center}
  \rule[-.5cm]{0pt}{8cm}
  \begin{pspicture}(-3,-4)(7,3)\psgrid
    \input{Exemples/interCC_bis_in}
  \end{pspicture}
\end{center}

\verbatiminput{Exemples/interCC_bis_in}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Examples gallery}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Basic geometry}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Drawing of the bissector}
    \nopagebreak[4]

\tabex{gal_biss}


\cbstart
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{German right angle}

In Germany, there is another convention for the right angle (remark
of U. Dirr).
\nopagebreak[4]

\tabex{german_ra}

\cbend
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Triangle lines}

\begin{center}
\psset{unit=2cm}
\input{Exemples/remarq}
\end{center}\nopagebreak[4]

\verbatiminput{Exemples/remarq_in}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Euler circle}

\begin{center}
\psset{unit=2cm}
\input{Exemples/euler}
\end{center}\nopagebreak[4]

\verbatiminput{Exemples/euler_in}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Orthocenter and hyperbola}

The orthocenter of a triangle whose points are on the branches of the
hyperbola ${\mathscr H} : y=a/x$ belong to this hyperbola.
\nopagebreak[4]

\begin{center}
\psset{unit=.5cm}
\input{Exemples/orthoethyper}
\end{center}\nopagebreak[4]

\verbatiminput{Exemples/orthoethyper_in}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{17 sides regular polygon}

Striking picture created by K. F. Gauss.
he also demonstrates that it is possible to build the polygons which
have $2^{2^p}+1$ sides, the following one has 257 sides!
\nopagebreak[4]

%% Polygone à 17 côtés
\bgroup\centering
\psset{unit=1.5cm, CodeFig=true, RightAngleSize=.14, CodeFigColor=red,
  CodeFigB=true, linestyle=dashed, dash=2mm 2mm}
\begin{pspicture}(-5.5,-5.5)(5.5,6)%\psgrid
  \pstGeonode[PosAngle=-90](0,0){O}%
  %% picture correct for x>0 and y<=0 for P_1!! 
  \pstGeonode[PosAngle=0](5,0){P_1}%
  \pstCircleOA{O}{P_1}%
  \pstSymO[PointSymbol=none, CodeFig=false]{O}{P_1}{PP_1}%
  \ncline[linestyle=solid]{PP_1}{P_1}%
  \pstRotation[RotAngle=90, PosAngle=90]{O}{P_1}{B}%
  \pstRightAngle{B}{O}{PP_1}\ncline[linestyle=solid]{O}{B}%
  \pstHomO[HomCoef=.25]{O}{B}{J}{J}%
  \ncline{J}{P_1}%
  \pstBissectBAC[PointSymbol=none]{O}{J}{P_1}{PE1}%
  \pstBissectBAC[PointSymbol=none]{O}{J}{PE1}{PE2}%
  \pstInterLL[PosAngle=-90]{O}{P_1}{J}{PE2}{E}%
  \pstRotation[PosAngle=-90, RotAngle=-45, PointSymbol=none]{J}{E}{PF1}%
  \pstInterLL[PosAngle=-90]{O}{P_1}{J}{PF1}{F}%
  \pstMiddleAB[PointSymbol=none]{F}{P_1}{MFP1}
  \pstCircleOA{MFP1}{P_1}%
  \pstInterLC[PointSymbolB=none]{O}{B}{MFP1}{P_1}{K}{H}%
  \pstCircleOA{E}{K}%
  \pstInterLC{O}{P_1}{E}{K}{N_4}{N_6}%
  \pstRotation[RotAngle=90, PointSymbol=none]{N_6}{E}{PP_6}{}%
  \pstInterLC[PosAngleA=90, PosAngleB=-90, PointNameB=P_{13}]%
    {N_6}{PP_6}{O}{P_1}{P_6}{P_13}%
  \pstSegmentMark[SegmentSymbol=wedge]{N_6}{P_6}%
  \pstSegmentMark[SegmentSymbol=wedge]{P_13}{N_6}%
  \pstRotation[RotAngle=90, PointSymbol=none]{N_4}{E}{PP_4}{}%
  \pstInterLC[PosAngleA=90, PosAngleB=-90, PointNameB=P_{15}]%
    {N_4}{PP_4}{O}{P_1}{P_4}{P_15}%
  \pstSegmentMark[SegmentSymbol=cup]{N_4}{P_4}%
  \pstSegmentMark[SegmentSymbol=cup]{P_15}{N_4}%
  \pstRightAngle{P_1}{N_6}{P_6}\pstRightAngle{P_1}{N_4}{P_4}%
  \pstBissectBAC[PosAngle=90, linestyle=none]{P_4}{O}{P_6}{P_5}%
  \pstOrtSym[PosAngle=-90, PointName=P_{14}]{O}{P_1}{P_5}{P_14}%
  \pstInterCC[PosAngleB=90, PointSymbolA=none,
    PointNameA=none]{O}{P_1}{P_4}{P_5}{H}{P_3}%
  \pstOrtSym[PosAngle=-90, SegmentSymbol=pstslash, PointName=P_{16}]%
    {O}{P_1}{P_3}{P_16}%
  \pstInterCC[PosAngleB=90, PointSymbolA=none]{O}{P_1}{P_3}{P_4}{H}{P_2}%
  \pstOrtSym[PosAngle=-90, SegmentSymbol=pstslashslashslash,
             PointName=P_{17}]{O}{P_1}{P_2}{P_17}%
  \pstInterCC[PosAngleA=90, PointSymbolB=none]{O}{P_1}{P_6}{P_5}{P_7}{H}
  \pstOrtSym[PosAngle=-90, SegmentSymbol=circ,
             PointName=P_{12}]{O}{P_1}{P_7}{P_12}%
  \pstInterCC[PosAngleA=100, PointSymbolB=none]{O}{P_1}{P_7}{P_6}{P_8}{H}
  \pstOrtSym[PosAngle=-100, SegmentSymbol=times,
             PointName=P_{11}]{O}{P_1}{P_8}{P_11}%
  \pstInterCC[PosAngleA=135, PointSymbolB=none]{O}{P_1}{P_8}{P_7}{P_9}{H}
  \pstOrtSym[PosAngle=-135, SegmentSymbol=equiv,
             PointName=P_{10}]{O}{P_1}{P_9}{P_10}%
  \psline[linecolor=green, linestyle=solid]%
    (P_1)(P_2)(P_3)(P_4)(P_5)(P_6)(P_7)(P_8)(P_9)%
         (P_10)(P_11)(P_12)(P_13)(P_14)(P_15)(P_16)(P_17)(P_1)%
\end{pspicture}
\egroup

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Circles tangents}

The drawing of the circle tangents which crosses a given point.
\nopagebreak[4]

\begin{pspicture}(15,9)%\psgrid
  %%% tangente à un cercle passant par un point
  \pstGeonode(5, 5){O}%
  \pstCircleOA[Radius=\pstDistVal{4}]{O}{}%
  \pstGeonode(14,2){M}%
  \pstMiddleAB[PointSymbol=none]{O}{M}{O'}
  \pstInterCC[RadiusA=\pstDistVal{4}, DiameterB=\pstDistAB{O}{M}]%
    {O}{}{O'}{}{A}{B}%
  \psset{linecolor=red, linewidth=1.3pt, nodesep=-2}
  \pstLineAB{M}{A}\pstLineAB{M}{B}
\end{pspicture}

The drawing of the common tangent of two circles.
\nopagebreak[4]

\begin{pspicture}(-2,0)(13,9)%\psgrid
  %% tangente à deux cercles
  \pstGeonode(9,3){O}\pstGeonode(3,6){O'}\psset{PointSymbol=none}%
  \pstCircleOA[Radius=\pstDistVal{3}]{O}{}\pstCircleOA[Radius=\pstDistVal{1}]{O'}{}
  \pstInterLC[Radius=\pstDistVal{3}]{O}{O'}{O}{}{M}{toto}
  \pstInterLC[Radius=\pstDistVal{1}]{O}{O'}{O'}{}{M'}{toto}
  \pstRotation[RotAngle=30]{O}{M}{N}%
  \pstRotation[RotAngle=30]{O'}{M'}{N'}%
  \pstInterLL[PointName=\Omega]{O}{O'}{N}{N'}{Omega}
  \pstMiddleAB[PointSymbol=none]{O}{Omega}{I}
  \pstInterCC{I}{O}{O}{M}{A}{B}
  \psset{nodesepA=-1, nodesepB=-3, linecolor=blue, linewidth=1.3pt}
  \pstLineAB[nodesep=-2]{A}{Omega}\pstLineAB[nodesep=-2]{B}{Omega}
  \pstRotation[RotAngle=-150]{O'}{M'}{N''}%
  \pstInterLL[PointName=\Omega']{O}{O'}{N}{N''}{Omega'}
  \pstMiddleAB[PointSymbol=none]{O}{Omega'}{J}
  \pstInterCC{J}{O}{O}{M}{A'}{B'}
  \psset{nodesepA=-1, nodesepB=-3, linecolor=red}
  \pstLineAB{A'}{Omega'}\pstLineAB{B'}{Omega'}
\end{pspicture}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Fermat's point}

Drawing of Manuel Luque.\nopagebreak[4]

\begin{pspicture}(-7,-6)(5,5)%\psgrid
  \psset{PointSymbol=none}
  \pstTriangle[PosAngleA=-160, PosAngleB=90, PosAngleC=-25]%
    (-3,-2){B}(0,3){A}(2,-1){C}%
  \psset{RotAngle=-60}
  \pstRotation[PosAngle=-90]{B}{C}{A'}
  \pstRotation{C}{A}{B'}
  \pstRotation[PosAngle=160]{A}{B}{C'}
  \pstLineAB{A}{B'}
  \pstLineAB{C}{B'}
  \pstLineAB{B}{A'}
  \pstLineAB{C}{A'}
  \pstLineAB{B}{C'}
  \pstLineAB{A}{C'}
  \pstCircleABC[linecolor=red]{A}{B}{C'}{O_1}
  \pstCircleABC[linecolor=blue]{A}{C}{B'}{O_2}
  \pstCircleABC[linecolor=Aquamarine]{A'}{C}{B}{O_3}
  \pstInterCC[PointSymbolA=none]{O_1}{A}{O_2}{A}{E}{F}
\end{pspicture}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Escribed and inscribed circles of a triangle}

%% cercles inscrit et exinscrits d'un triangle
\bgroup\centering
\psset{unit=1cm, dash=5mm 4mm, PointSymbolA=none, PointSymbolB=none}
\begin{pspicture}(-6,-5)(11,15)%\psgrid
  \psframe(-6,-5)(11,15)
  \pstTriangle[linewidth=2pt, PosAngleA=-75, PosAngleB=180,
               PosAngleC=45, linecolor=red]%
    (4,1){A}(0,3){B}(5,5){C}%
  \psset{linecolor=blue}
  \pstBissectBAC[PointSymbol=none]{C}{A}{B}{AB}
  \pstBissectBAC[PointSymbol=none]{A}{B}{C}{BB}
  \pstBissectBAC[PointSymbol=none]{B}{C}{A}{CB}
  \pstInterLL{A}{AB}{B}{BB}{I}
  \psset{linecolor=magenta, linestyle=dashed}
  \pstProjection[PosAngle=-90]{A}{B}{I}{I_C}
  \pstLineAB{I}{I_C}\pstRightAngle[linestyle=solid]{A}{I_C}{I}
  \pstProjection{A}{C}{I}{I_B}
  \pstLineAB{I}{I_B}\pstRightAngle[linestyle=solid]{C}{I_B}{I}
  \pstProjection[PosAngle=80]{C}{B}{I}{I_A}
  \pstLineAB{I}{IA}\pstRightAngle[linestyle=solid]{B}{I_A}{I}
  \pstCircleOA[linecolor=yellow, linestyle=solid]{I}{I_A}
  %% BISSECTRICES EXTÉRIEURES
  \psset{linecolor=magenta, linestyle=none}
  \pstOutBissectBAC[PointSymbol=none]{C}{A}{B}{AOB}
  \pstOutBissectBAC[PointSymbol=none]{A}{B}{C}{BOB}
  \pstOutBissectBAC[PointSymbol=none]{B}{C}{A}{COB}
  \pstInterLL[PosAngle=-90]{A}{AOB}{B}{BOB}{I_1}
  \pstInterLL{A}{AOB}{C}{COB}{I_2}
  \pstInterLL[PosAngle=90]{C}{COB}{B}{BOB}{I_3}
  \psset{linecolor=magenta, linestyle=dashed}
  \pstProjection[PosAngle=50, PointName=I_{1C}]{A}{B}{I_1}{I1C}
  \pstLineAB{I_1}{I1C}\pstRightAngle[linestyle=solid]{I_1}{I1C}{A}
  \pstProjection[PointName=I_{1B}]{A}{C}{I_1}{I1B}
  \pstLineAB{I_1}{I1B}\pstRightAngle[linestyle=solid]{A}{I1B}{I_1}
  \pstProjection[PosAngle=110, PointName=I_{1A}]{C}{B}{I_1}{I1A}
  \pstLineAB{I_1}{I1A}\pstRightAngle[linestyle=solid]{I_1}{I1A}{C}
  \pstProjection[PointName=I_{2B}]{A}{C}{I_2}{I2B}
  \pstLineAB{I_2}{I2B}\pstRightAngle[linestyle=solid]{A}{I2B}{I_2}
  \pstProjection[PosAngle=-90, PointName=I_{2C}]{A}{B}{I_2}{I2C}
  \pstLineAB{I_2}{I2C}\pstRightAngle[linestyle=solid]{I_2}{I2C}{A}
  \pstProjection[PosAngle=90, PointName=I_{2A}]{B}{C}{I_2}{I2A}
  \pstLineAB{I_2}{I2A}\pstRightAngle[linestyle=solid]{C}{I2A}{I_2}
  \pstProjection[PosAngle=130, PointName=I_{3A}]{C}{B}{I_3}{I3A}
  \pstLineAB{I_3}{I3A}\pstRightAngle[linestyle=solid]{C}{I3A}{I_3}
  \pstProjection[PosAngle=-90, PointName=I_{3C}]{A}{B}{I_3}{I3C}
  \pstLineAB{I_3}{I3C}\pstRightAngle[linestyle=solid]{A}{I3C}{I_3}
  \pstProjection[PointName=I_{3B}]{C}{A}{I_3}{I3B}
  \pstLineAB{I_3}{I3B}\pstRightAngle[linestyle=solid]{I_3}{I3B}{A}
  \psset{linecolor=yellow, linestyle=solid}
  \pstCircleOA{I_1}{I1C}
  \pstCircleOA{I_2}{I2B}
  \pstCircleOA{I_3}{I3A}
  \psset{linecolor=red, linestyle=solid, nodesepA=-1, nodesepB=-1}
  \pstLineAB{I1B}{I3B}\pstLineAB{I1A}{I2A}\pstLineAB{I2C}{I3C}
\end{pspicture}
\egroup

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Some locus points}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Parabola}

\begin{minipage}[m]{.33\linewidth}
The parabola is the set of points which is at the same distance
between a point and a line.
\end{minipage}
\newcommand{\NbPt}{11}
\input{Exemples/parabole}\nopagebreak[4]

\verbatiminput{Exemples/parabole_in}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Hyperbola}

\begin{minipage}[m]{.33\linewidth}
The hyperbola is the set of points whose difference between their
distance of two points (the focus) is constant.
\end{minipage}
%% QQ DEFINITIONS
\newcommand{\Sommet}{1.4142135623}\newcommand{\PosFoyer}{2}
\newcommand{\HypAngle}{0}
\setcounter{i}{0}\newcounter{CoefDiv}\setcounter{CoefDiv}{20}
\newcounter{Inc}\setcounter{Inc}{1}\newcounter{n}\setcounter{n}{1}
\newcommand{\Ri}{%
  \PosFoyer\space\Sommet\space%
  sub \arabic{i}\space\arabic{CoefDiv}\space%
  div add}
\newcommand{\Rii}{\Ri\space \Sommet\space 2 mul add .001 add}
\begin{pspicture}[.5](-4,-4)(4,4)%\psgrid
  \pstGeonode[PosAngle=90](0,0){O}%
  \pstGeonode(\PosFoyer;\HypAngle){F}%
  \pstSymO[PosAngle=180]{O}{F}{F'}%
  \pstLineAB{F}{F'}
  %% TRACÉ DES ASYMPTOTES
  %\psset{PointSymbol=none}
  \pstCircleOA{O}{F}
  \pstGeonode[PosAngle=-135](\Sommet;\HypAngle){S}
  \pstRotation[RotAngle=90, PointSymbol=none]{S}{O}{B}
  \pstInterLC[PosAngleA=90, PosAngleB=-90]{S}{B}{O}{F}{A_1}{A_2}
  \pstLineAB[nodesepA=-3,nodesepB=-5]{A_1}{O}
  \pstLineAB[nodesepA=-3,nodesepB=-5]{A_2}{O}
  \pstMarkAngle[LabelSep=.8, MarkAngleRadius=.7, arrows=->]{F}{O}{A_1}{$\Psi$}
  \ncline[linecolor=red]{A_1}{A_2}
  \pstRightAngle[RightAngleSize=.15]{A_1}{S}{O}
  \psset{PointName=none}
  \whiledo{\value{n}<8}{%
    \psset{RadiusA=\pstDistVal{\Ri},RadiusB=\pstDistVal{\Rii},PointSymbol=none}
    \pstInterCC{F}{}{F'}{}{M\arabic{n}}{P\arabic{n}}
    \pstInterCC{F'}{}{F}{}{M'\arabic{n}}{P'\arabic{n}}
    \stepcounter{n}\addtocounter{i}{\value{Inc}}\addtocounter{Inc}{\value{Inc}}
  } %% fin de whiledo
  \psset{linecolor=blue}
  \pstGenericCurve[GenCurvFirst=P1]{M}{1}{7}\pstGenericCurve{P}{1}{7}
  \pstGenericCurve[GenCurvFirst=P'1]{M'}{1}{7}\pstGenericCurve{P'}{1}{7}
\end{pspicture}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Cycloid}
    
The wheel rolls from $M$ to $A$. The circle points are on a
cycloids.\nopagebreak[4]

\begin{center}
\input{Exemples/cyclo}
\end{center}\nopagebreak[4]

\verbatiminput{Exemples/cyclo_in}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Hypocycloids (Astroid and Deltoid)}

A wheel rolls inside a circle, and depending of the radius ratio, it
is an astroid, a deltoid and in the general case hypo-cycloids.
\nopagebreak[4]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% ASTROIDE
\input{Exemples/hypocyclo}
%%%%%%%%%%%%%%%%%%%%
\begin{center}
\input{Exemples/astro}\input{Exemples/delto}
\end{center}

\verbatiminput{Exemples/hypocyclo}
\verbatiminput{Exemples/astro_in}

\cbstart
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Lines and circles envelope}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Conics}

Let's consider a circle and a point $A$ not on the circle. The
set of all the mediator lines of segments defined by $A$ and the
circle points, create two conics depending of the position of $A$:

\begin{itemize}
\item inside the circle: an hyperbola;
\item outside the circle: an ellipse.
\end{itemize}

(figure of O. Reboux).

\begin{center}\input{Exemples/envellipse}\end{center}

\verbatiminput{Exemples/envellipse_in}

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \subsubsection{Cardioid}

The cardioid is defined by the circles centred on a circle and
crossing a given point.

\begin{center}\input{Exemples/envcardi}\end{center}

\verbatiminput{Exemples/envcardi_in}

\cbend
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Homotethy and fractals}

\tabex{fracthom}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{hyperbolic geometry: a triangle and its altitudes}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Tracé de géodésique en géométrie hyperbolique
%% Attention ne fonctionne que si les points ne sont pas alignés avec O
%% Ceci est un cas particulier, je ne crois pas que les hauteurs
%% soient concourantes pour tous les triangles hyperboliques.
\begin{pspicture}(-5,-5)(5,5)
  \psclip{\pscircle(0,0){4}}%\psgrid
    %\newlength{\radius}\setlength{\radius}{0cm}
    %\newcounter{rapport}\setcounter{i}{1}
    %\whiledo{\value{i}<100}{
    %  \setlength{\radius}{4cm*\value{i}}
    %  \setcounter{rapport}{\value{i}+1}
    %  \divide\radius by \arabic{rapport}
    %  \pscircle[linestyle=dotted, linecolor=gray]%
    %    (0, 0){\radius}
    %  \setcounter{i}{\value{i}*2}
    %  }
    \pstGeonode(1, 2){M}\pstGeonode(-2,2){N}\pstGeonode(0,-2){P}%
    \psset{DrawCirABC=false, PointSymbol=none}%
    \pstGeonode(0,0){O}\pstGeonode(4,0){A}\pstCircleOA{O}{A}%
    \pstHomO[HomCoef=\pstDistAB{O}{A} 2 mul \pstDistAB{O}{M} sub
      \pstDistAB{O}{M} div]{O}{M}{M'}%
    \pstHomO[HomCoef=\pstDistAB{O}{A} 2 mul \pstDistAB{O}{P} sub
      \pstDistAB{O}{P} div]{O}{P}{P'}%
    \pstHomO[HomCoef=\pstDistAB{O}{A} 2 mul \pstDistAB{O}{N} sub
      \pstDistAB{O}{N} div]{O}{N}{N'}%
    \psset{linecolor=green, linewidth=1.5pt}%
    \pstCircleABC{M}{N}{M'}{OmegaMN}\pstArcOAB{OmegaMN}{N}{M}%
    \pstCircleABC{M}{P}{M'}{OmegaMP}\pstArcOAB{OmegaMP}{M}{P}%
    \pstCircleABC{N}{P}{P'}{OmegaNP}\pstArcOAB{OmegaNP}{P}{N}%
    \psset{linecolor=blue}
    %% la hauteur issue de M
    \pstHomO[HomCoef=\pstDistAB{OmegaNP}{N} 2 mul \pstDistAB{OmegaNP}{M} sub
      \pstDistAB{OmegaNP}{M} div]{OmegaNP}{M}{MH'}
    \pstCircleABC{M}{M'}{MH'}{OmegaMH}\pstArcOAB{OmegaMH}{MH'}{M}
    %% la hauteur issue de N
    \pstHomO[HomCoef=\pstDistAB{OmegaMP}{M} 2 mul \pstDistAB{OmegaMP}{N} sub
      \pstDistAB{OmegaMP}{N} div]{OmegaMP}{N}{NH'}
    \pstCircleABC{N}{N'}{NH'}{OmegaNH}\pstArcOAB{OmegaNH}{N}{NH'}
    %% la hauteur issue de P
    \pstHomO[HomCoef=\pstDistAB{OmegaMN}{M} 2 mul \pstDistAB{OmegaMN}{P} sub
      \pstDistAB{OmegaMN}{P} div]{OmegaMN}{P}{PH'}
    \pstCircleABC{P}{P'}{PH'}{OmegaPH}\pstArcOAB{OmegaPH}{P}{PH'}
  \endpsclip
\end{pspicture}

\end{document}

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \subsection{Une figure fractale : le flocon de \textsc{von Koch}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% TENTATIVE DE FLOCON DE VON KOCH
\newcounter{nbvk}%
%%%%%%%%%%%%%
%% PILE LIFO%
\newcounter{lifon}\setcounter{lifon}{1}%%
\newcommand{\Push}[1]{%%
  \expandafter\edef\csname lifocmd\roman{lifon}\endcsname{#1}%%
  \addtocounter{lifon}{1}%%
  }%
\newcommand{\Pop}{\csname lifocmd\roman{lifon}\endcsname}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcounter{nombre}\setcounter{nombre}{0}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\vonkoch}[3]{%%
  \setcounter{nbvk}{#1}%%
  \edef\Pointi{#2}\edef\Pointii{#3}%%
  \ifnum\value{nbvk}=1%
    \ncline{\Pointi}{\Pointii}%
  \else % ELSE%
    \addtocounter{nombre}{1}%
    %% position des nouveauX points%
    \pstHomO[HomCoef=1 3 div]{\Pointi}{\Pointii}{P\arabic{nombre}1}%
    \pstHomO[HomCoef=2 3 div]{\Pointi}{\Pointii}{P\arabic{nombre}2}%
    \pstRotation[RotAngle=60]{P\arabic{nombre}1}{P\arabic{nombre}2}{P\arabic{nombre}3}%
    \addtocounter{nbvk}{-1}%%
    \Push{\Pointi}\Push{\Pointii}\Push{\arabic{nombre}}\Push{\arabic{nbvk}}%
    \vonkoch{\value{nbvk}}{\Pointi}{P\arabic{nombre}1}%
    \addtocounter{lifon}{-1}\edef\nbvklocal{\Pop}%
    \addtocounter{lifon}{-1}\edef\nombrelocal{\Pop}%
    \addtocounter{lifon}{-1}\edef\Pointii{\Pop}%
    \addtocounter{lifon}{-1}\edef\Pointi{\Pop}%
    \addtocounter{lifon}{4}%
    \vonkoch{\nbvklocal}{P\nombrelocal1}{P\nombrelocal3}%
    \addtocounter{lifon}{-1}\edef\nbvklocal{\Pop}%
    \addtocounter{lifon}{-1}\edef\nombrelocal{\Pop}%
    \addtocounter{lifon}{-1}\edef\Pointii{\Pop}%
    \addtocounter{lifon}{-1}\edef\Pointi{\Pop}%
    \addtocounter{lifon}{4}%
    \vonkoch{\nbvklocal}{P\nombrelocal3}{P\nombrelocal2}%
    \addtocounter{lifon}{-1}\edef\nbvklocal{\Pop}%
    \addtocounter{lifon}{-1}\edef\nombrelocal{\Pop}%
    \addtocounter{lifon}{-1}\edef\Pointii{\Pop}%
    \addtocounter{lifon}{-1}\edef\Pointi{\Pop}%
    \addtocounter{lifon}{4}%
    \vonkoch{\nbvklocal}{P\nombrelocal2}{\Pointii}%
    \addtocounter{lifon}{-4}%
  \fi%
}%
\psset{unit=2.5cm}%
\begin{pspicture}(-2,-2)(2,2)%
  \psset{PointSymbol=none}
  \pstGeonode(-2,-2){A}\pstGeonode(2,2){B}%
  \pstRotation[RotAngle=60]{B}{A}{C}%
  \vonkoch{5}{A}{B}\setcounter{nombre}{0}%%%
  \vonkoch{4}{B}{C}\setcounter{nombre}{0}%%%
  \vonkoch{4}{C}{A}%%
\end{pspicture}%
%
\end{document}%

\begin{pspicture*}(-4,-5)(4,5)%\psgrid
  \pstGeonode[PosAngle=-135](0,0){O}
  \pstGeonode[PosAngle=-90](1,0.5){I}
  \pstGeonode[PosAngle=-180](0.5,2){J}
  \pstLineAB[nodesep=100]{O}{I}
  \pstLineAB[nodesep=100]{O}{J}
  \multips(-5,-2.5)(1,0.5){11}{\psline(0,-.15)(0,.15)}%
  \multips(-5,-2.5)(1,0.5){11}{\psline[linestyle=dotted](-10,-40)(10,40)}%
  \multips(-2,-8)(0.5,2){9}{\psline(-.15,0)(.15,0)}%
  \multips(-2,-8)(0.5,2){9}{\psline[linestyle=dotted](-10,-5)(10,5)}%
  %% huit points
  \psset{PointSymbol=x}
  \pstOIJGeonode(1,2){A}{O}{I}{J}
  \pstOIJGeonode(-2,1){B}{O}{I}{J}
  \pstOIJGeonode(-1,-1.5){C}{O}{I}{J}
  \pstOIJGeonode(2,-1){D}{O}{I}{J}
\end{pspicture*}
%