%@Auteur: François Meria\par \begin{multicols}{2} \begin{center} \psset{unit=0.20cm} \pspicture(-11,-11)(11,7) \pstGeonode[PointSymbol=none,PosAngle={-135,135,45,135},PointName=none](0,0){O}(-10,0){D}(10,0){D'}(0,10){A} \pstCircleOA[linestyle=dashed]{O}{D} \pstGeonode[PointName=none,PointSymbol=none](0,-10){Z} \pstGeonode[PointSymbol=none,PosAngle=45,PointName=none](5,0){B} \pstGeonode[PointName=none,PointSymbol=none](5,11){T}(5,-11){T'} \pstInterLC[PointSymbol=none,PointName=none,PosAngle=45]{D}{D'}{B}{A}{E}{E'} \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{A}{M_1}{-80} \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{A}{M_2}{320} \pstCurvAbsNode[PointName=none,PointSymbol=none]{A}{E}{K_1}{-150} \pstCurvAbsNode[PointName=none,PointSymbol=none]{A}{E}{K_2}{430} \pstInterCC[PointSymbol=none,PosAngleA=180,PointName=none]{O}{D}{A}{E}{A_1}{A_4} \pstInterCC[PointSymbol=none,PointName=none,PosAngle=225]{O}{D}{A_1}{A}{A_2}{H} \pstInterCC[PointSymbol=none,PointName=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} \pstCurvAbsNode[PointName=none,PointSymbol=none]{A_4}{A_3}{Q_2}{25} \pspolygon[fillstyle=solid,fillcolor=lightgray,linewidth=0.5mm](A)(A_2)(A_4)(A_1)(A_3) \endpspicture \end{center} \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} \psset{unit=0.20cm} \pspicture(-11,-11)(11,11) \pstGeonode[PointSymbol=x,PosAngle={-90,135,45}](0,0){O}(-10,0){D}(10,0){D'} \pstCircleOA{O}{D} \pstLineAB[nodesep=-1]{D}{D'} \endpspicture \vskip 0.3cm \textbf{\'Etape 1.} \end{center} %% figure 2 \begin{center} \psset{unit=0.20cm} \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} \psset{unit=0.20cm} \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} \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=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} \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} \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} \psset{unit=0.20cm} \pspicture(-11,-10)(11,11) \pstGeonode[PointSymbol=x,PosAngle={-135,180,0,135}](0,0){O}(-10,0){D}(10,0){D'}(0,10){A} \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} \pstCurvAbsNode[PointName=none,PointSymbol=none]{A_4}{A_3}{Q_2}{25} \pspolygon[fillstyle=solid,fillcolor=lightgray,linewidth=0.5mm](A)(A_2)(A_4)(A_1)(A_3) \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}