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%@Auteur: François Meria\par
\begin{multicols}{2}
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\columnbreak
 
L'étoile ci-contre s'appelle un \textit{pentagramme}.\\
On veut réaliser la construction d'un pentagramme en partant d'un
cercle de rayon 8~cm.\\
Pour cela, la réalisation passera par les sept étapes de
construction décrites plus bas.
\end{multicols}
 
\textbf{\'Etapes de la construction}.
 
\begin{multicols}{3}\setlength{\columnseprule}{0.5pt}
%figure 1
\begin{center}
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        \pstCircleOA{O}{D}
        \pstLineAB[nodesep=-1]{D}{D'}
    \endpspicture
 
\vskip 0.3cm
 
\textbf{\'Etape 1.}
\end{center}
 
%% figure 2
\begin{center}
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    \pspicture(-11,-11)(11,11)
        \pstGeonode[PointSymbol=x,PosAngle={-135,135,45,135}](0,0){O}(-10,0){D}(10,0){D'}(0,10){A}
        \pstCircleOA{O}{D}
        \pstLineAB[nodesep=-1]{D}{D'}
        \pstGeonode[PointName=none,PointSymbol=none](0,-10){Z}
        \pstLineAB[nodesep=-1.5]{A}{Z}
        \pstRightAngle[RightAngleSize=1]{A}{O}{D'}
        \pstSegmentMark[SegmentSymbol=pstslashh]{D}{O}
        \pstSegmentMark[SegmentSymbol=pstslashh]{O}{D'}
    \endpspicture
 
\vskip 0.3cm
 
\textbf{\'Etape 2.}
\end{center}
 
%%% figure 3
\begin{center}
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    \pspicture(-11,-11)(11,11)
        \pstGeonode[PointSymbol=x,PosAngle={-135,135,45,135}](0,0){O}(-10,0){D}(10,0){D'}(0,10){A}
        \pstCircleOA{O}{D}
        \pstLineAB[nodesep=-1]{D}{D'}
        \pstGeonode[PointName=none,PointSymbol=none](0,-10){Z}
        \pstLineAB[nodesep=-1.5]{A}{Z}
        \pstGeonode[PointSymbol=x,PosAngle=225](5,0){B}
        \pstGeonode[PointName=none,PointSymbol=none](5,11){T}(5,-11){T'}
        \pstLineAB[nodesep=-1.5]{T}{T'}
        \pstLineAB[linestyle=dashed]{A}{B}
        \pstRightAngle[RightAngleSize=1]{T}{B}{D'}
        \pstSegmentMark[SegmentSymbol=pstslashhh]{O}{B}
        \pstSegmentMark[SegmentSymbol=pstslashhh]{B}{D'}
    \endpspicture
 
\vskip 0.3cm
 
\textbf{\'Etape 3.}
\end{center}
\end{multicols}
 
\begin{multicols}{3}
%%%% figure 4
\begin{center}
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    \pspicture(-11,-10)(10,11)
        \pstGeonode[PointSymbol=x,PosAngle={-135,135,45,135}](0,0){O}(-10,0){D}(10,0){D'}(0,10){A}
        \pstCircleOA{O}{D}
        \pstLineAB[nodesep=-1]{D}{D'}
        \pstGeonode[PointName=none,PointSymbol=none](0,-10){Z}
        \pstLineAB[nodesep=-1.5]{A}{Z}
        \pstGeonode[PointSymbol=x,PosAngle=45](5,0){B}
        \pstGeonode[PointName=none,PointSymbol=none](5,11){T}(5,-11){T'}
        \pstLineAB[nodesep=-1.5]{T}{T'}
        \pstLineAB[linestyle=dashed]{A}{B}
        \pstInterLC[PointSymbol=none,PointNameB=none,PosAngle=135]{D}{D'}{B}{A}{E}{E'}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{A}{M_1}{-20}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{A}{M_2}{160}
        \pstArcOAB{B}{M_1}{M_2}
    \endpspicture
 
\vskip 0.3cm
 
\textbf{\'Etape 4.}
\end{center}
 
%%%%% figure 5
\begin{center}
\psset{unit=0.20cm}
    \pspicture(-11,-10)(10,11)
        \pstGeonode[PointSymbol=x,PosAngle={-135,135,45,135}](0,0){O}(-10,0){D}(10,0){D'}(0,10){A}
        \pstCircleOA{O}{D}
        \pstLineAB[nodesep=-1]{D}{D'}
        \pstGeonode[PointName=none,PointSymbol=none](0,-10){Z}
        \pstLineAB[nodesep=-1.5]{A}{Z}
        \pstGeonode[PointSymbol=x,PosAngle=45](5,0){B}
        \pstGeonode[PointName=none,PointSymbol=none](5,11){T}(5,-11){T'}
        \pstLineAB[nodesep=-1.5]{T}{T'}
        \pstLineAB[linestyle=dashed]{A}{B}
        \pstInterLC[PointSymbol=none,PointNameB=none,PosAngle=45]{D}{D'}{B}{A}{E}{E'}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{A}{M_1}{-20}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{A}{M_2}{160}
        \pstArcOAB{B}{M_1}{M_2}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A}{E}{K_1}{-35}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A}{E}{K_2}{110}
        \pstArcOAB{A}{K_1}{K_2}
        \pstInterCC[PointSymbol=none,PosAngleA=180]{O}{D}{A}{E}{A_1}{A_4}
    \endpspicture
 
\vskip 0.3cm
 
\textbf{\'Etape 5.}
\end{center}
 
%%%%%% figure 6
\begin{center}
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    \pspicture(-11,-10)(10,11)
        \pstGeonode[PointSymbol=x,PosAngle={-135,135,45,135}](0,0){O}(-10,0){D}(10,0){D'}(0,10){A}
        \pstCircleOA{O}{D}
        \pstLineAB[nodesep=-1]{D}{D'}
        \pstGeonode[PointName=none,PointSymbol=none](0,-10){Z}
        \pstLineAB[nodesep=-1.5]{A}{Z}
        \pstGeonode[PointSymbol=x,PosAngle=45](5,0){B}
        \pstGeonode[PointName=none,PointSymbol=none](5,11){T}(5,-11){T'}
        \pstLineAB[nodesep=-1.5]{T}{T'}
        \pstLineAB[linestyle=dashed]{A}{B}
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        \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{A}{M_1}{-20}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{A}{M_2}{160}
        \pstArcOAB{B}{M_1}{M_2}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A}{E}{K_1}{-35}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A}{E}{K_2}{110}
        \pstArcOAB{A}{K_1}{K_2}
        \pstInterCC[PointSymbol=none,PosAngleA=180]{O}{D}{A}{E}{A_1}{A_4}
        \pstInterCC[PointSymbol=x,PointNameB=none,PosAngle=225]{O}{D}{A_1}{A}{A_2}{H}
        \pstInterCC[PointSymbol=x,PointNameA=none,PosAngle=-45]{O}{D}{A_4}{A}{R}{A_3}
 
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A_1}{A_2}{P_1}{-35}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A_1}{A_2}{P_2}{25}
        \pstArcOAB{A_1}{P_1}{P_2}
 
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A_4}{A_3}{Q_1}{-35}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A_4}{A_3}{Q_2}{25}
        \pstArcOAB{A_4}{Q_1}{Q_2}
    \endpspicture
 
\vskip 0.3cm
 
\textbf{\'Etape 6.}
\end{center}
\end{multicols}
\setlength{\columnseprule}{0pt}
%%%%%%% figure 7
\begin{center}
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    \pspicture(-11,-10)(11,11)
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        \pstGeonode[PointName=none,PointSymbol=none](0,-10){Z}
        \pstGeonode[PointSymbol=x,PosAngle=45](5,0){B}
        \pstGeonode[PointName=none,PointSymbol=none](5,11){T}(5,-11){T'}
        \pstInterLC[PointSymbol=none,PointNameB=none,PosAngle=45]{D}{D'}{B}{A}{E}{E'}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{A}{M_1}{-20}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{A}{M_2}{160}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A}{E}{K_1}{-35}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A}{E}{K_2}{110}
        \pstInterCC[PointSymbol=none,PosAngleA=180]{O}{D}{A}{E}{A_1}{A_4}
        \pstInterCC[PointSymbol=x,PointNameB=none,PosAngle=225]{O}{D}{A_1}{A}{A_2}{H}
        \pstInterCC[PointSymbol=x,PointNameA=none,PosAngle=-45]{O}{D}{A_4}{A}{R}{A_3}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A_1}{A_2}{P_1}{-35}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A_1}{A_2}{P_2}{25}
        \pstCurvAbsNode[PointName=none,PointSymbol=none]{A_4}{A_3}{Q_1}{-35}
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        \pstCircleOA{O}{D}
        \pstLineAB[nodesep=-1]{D}{D'}
        \pstLineAB[nodesep=-1.5]{A}{Z}
        \pstLineAB[nodesep=-1.5]{T}{T'}
        \pstGeonode[PointSymbol=x,PosAngle=-45](5,0){B}
    \endpspicture
 
\vskip 0.3cm
 
\textbf{\'Etape 7.}
\end{center}