\documentclass[a4paper,11pt]{article} \usepackage{francois_meria} \usepackage[dvips]{graphicx} \usepackage[dvips]{epsfig} \setlength{\parindent}{0mm} \lhead{\textsf{Collège Château Forbin} - \textit{Mathématiques} - \textsf{6\ieme}} \chead{} \rhead{\textit{Année} 2005/2006} \pagestyle{fancy} \renewcommand{\headrulewidth}{0.5pt} \begin{document} \begin{center} \begin{tabularx}{\textwidth}{|X|} \hline \vskip 0.1cm \begin{center} {\Large\textbf{Construction géométrique d'un Pentagramme}}\\ \vskip 0.025cm \end{center}\\ \hline \end{tabularx} \end{center} \vskip 0.55cm \begin{multicols}{2}\setlength{\columnseprule}{0.5pt} 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. \columnbreak \begin{center} \psset{unit=0.20cm} \pspicture(-11,-11)(11,7) 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\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} \end{multicols} \vskip 0.5cm \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} \vskip 0.5cm \begin{multicols}{3}\setlength{\columnseprule}{0.5pt} %%%% 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} \vskip 0.5cm %%%%%%% 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} \newpage \textbf{\LARGE Programme de construction }\\ \large{\texttt{Toutes les constructions doivent se faire au COMPAS et à la règle (sans utiliser les graduations sauf pour l'étape 1).}}\\ \vskip 2.5cm \shadowbox{ \begin{minipage}[c]{\textwidth} \textbf{\'Etape 1.}\\ Construire un cercle $\mathcal{C}$ de centre $O$ et de diamètre horizontal $[DD']$ tel que $DD'=16~$cm.\\ \textbf{\'Etape 2.}\\ Construire la médiatrice du segment $[DD']$. Nommer $A$ le point du cercle au Nord, c'est-à-dire sur le cercle, sur la médiatrice de $[DD']$ en haut du cercle.\\ \textbf{\'Etape 3.}\\ Construire la médiatrice du segment $[OD']$ et nommer $B$ le milieu du $[OD']$.\\ Tracer le segment $[AB]$ en pointillés.\\ \textbf{\'Etape 4.}\\ Tracer le cercle de centre $B$ passant par $A$ et nommer $E$ le point d'intersection du segment $[DO]$ et de ce cercle.\\ \textbf{\'Etape 5.}\\ Tracer le cercle de centre $A$ et de rayon $[AE]$. Il coupe le cercle $\mathcal{C}$ de départ en $A_1$ et $A_4$.\\ \textbf{\'Etape 6.}\\ Le cercle de centre $A_1$ passant par $A$ recoupe le cercle ${\cal C}$ en $A_2$.\\ Le cerlce de centre $A_4$ passant par $A$ recoupe le cercle ${\cal C}$ en $A_3$.\\ \textbf{\'Etape 7.}\\ Tracer le polygone $AA_2A_4A_1A_3$ puis effacer les traits de construction. Enfin colorier le \textit{pentagramme} ainsi obtenu. \end{minipage} } \end{document}