\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 : Quel polygone !}}\\ \vskip 0.025cm \end{center}\\ \hline \end{tabularx} \end{center} \vskip 0.55cm Il s'agit de construire la figure 2 ci-dessous. \begin{multicols}{2} \begin{center} \psset{unit=0.5cm} \pspicture(-9,-9)(9,9) \pstGeonode[PointSymbol=none,PosAngle={135,180,0,90,-90}](0,0){Q}(-8,0){N}(8,0){F}(0,8){B}(0,-8){J} \pstGeonode[PointSymbol=none,PointName=none](4,0){T} \pstCircleOA{Q}{F} \pstCircleOA[linestyle=dashed]{Q}{T} \pstBissectBAC[PointSymbol=none]{F}{Q}{B}{D} \pstBissectBAC[PointSymbol=none,PosAngle=135]{B}{Q}{N}{P} \pstBissectBAC[PointSymbol=none,PosAngle=-135]{N}{Q}{J}{L} \pstBissectBAC[PointSymbol=none]{J}{Q}{F}{H} \pstBissectBAC[PointName=none,PointSymbol=none]{F}{Q}{D}{M_1} \pstBissectBAC[PointName=none,PointSymbol=none]{D}{Q}{B}{M_2} \pstBissectBAC[PointName=none,PointSymbol=none]{B}{Q}{P}{M_3} \pstBissectBAC[PointName=none,PointSymbol=none]{P}{Q}{N}{M_4} \pstBissectBAC[PointName=none,PointSymbol=none]{N}{Q}{L}{M_5} \pstBissectBAC[PointName=none,PointSymbol=none]{L}{Q}{J}{M_6} \pstBissectBAC[PointName=none,PointSymbol=none]{J}{Q}{H}{M_7} \pstBissectBAC[PointName=none,PointSymbol=none]{H}{Q}{F}{M_8} \pstInterLC[PointSymbol=none,PosAngleA=-100,PosAngleB=55]{Q}{M_1}{Q}{T}{M}{E} \pstInterLC[PointSymbol=none,PosAngleA=-80,PosAngleB=42]{Q}{M_2}{Q}{T}{K}{C} \pstInterLC[PointSymbol=none,PosAngleA=-100,PosAngleB=50]{Q}{M_3}{Q}{T}{I}{A} \pstInterLC[PointSymbol=none,PosAngleA=-90,PosAngleB=90]{Q}{M_4}{Q}{T}{G}{O} \psline(N)(F) \psline(B)(J) \pstBissectBAC[PointSymbol=none,PointName=none]{F}{Q}{B}{D} \pstBissectBAC[PointSymbol=none,PointName=none]{B}{Q}{N}{P} \pstBissectBAC[PointSymbol=none,PointName=none]{N}{Q}{J}{L} \pstBissectBAC[PointSymbol=none,PointName=none]{J}{Q}{F}{H} \pstBissectBAC[PointName=none,PointSymbol=none]{F}{Q}{D}{M_1} \pstBissectBAC[PointName=none,PointSymbol=none]{D}{Q}{B}{M_2} \pstBissectBAC[PointName=none,PointSymbol=none]{B}{Q}{P}{M_3} \pstBissectBAC[PointName=none,PointSymbol=none]{P}{Q}{N}{M_4} \pstBissectBAC[PointName=none,PointSymbol=none]{N}{Q}{L}{M_5} \pstBissectBAC[PointName=none,PointSymbol=none]{L}{Q}{J}{M_6} \pstBissectBAC[PointName=none,PointSymbol=none]{J}{Q}{H}{M_7} \pstBissectBAC[PointName=none,PointSymbol=none]{H}{Q}{F}{M_8} \put(-1.5,-10){Figure 1.} \endpspicture \end{center} \columnbreak \begin{center} \psset{unit=0.5cm} \pspicture(-9,-9)(9,9) \pscircle[fillstyle=solid,fillcolor=lightgray](0,0){8} \pstGeonode[PointSymbol=none,PosAngle={-90,180,0,90,-90},PointName=none](0,0){Q}(-8,0){N}(8,0){F}(0,8){B}(0,-8){J} \pstGeonode[PointSymbol=none,PointName=none](4,0){T} \pstCircleOA{Q}{F} \pstCircleOA[linestyle=dashed]{Q}{T} \pstBissectBAC[PointSymbol=none,PointName=none]{F}{Q}{B}{D} \pstBissectBAC[PointSymbol=none,PointName=none]{B}{Q}{N}{P} \pstBissectBAC[PointSymbol=none,PointName=none]{N}{Q}{J}{L} \pstBissectBAC[PointSymbol=none,PointName=none]{J}{Q}{F}{H} \pstBissectBAC[PointName=none,PointSymbol=none]{F}{Q}{D}{M_1} \pstBissectBAC[PointName=none,PointSymbol=none]{D}{Q}{B}{M_2} \pstBissectBAC[PointName=none,PointSymbol=none]{B}{Q}{P}{M_3} \pstBissectBAC[PointName=none,PointSymbol=none]{P}{Q}{N}{M_4} \pstBissectBAC[PointName=none,PointSymbol=none]{N}{Q}{L}{M_5} \pstBissectBAC[PointName=none,PointSymbol=none]{L}{Q}{J}{M_6} \pstBissectBAC[PointName=none,PointSymbol=none]{J}{Q}{H}{M_7} \pstBissectBAC[PointName=none,PointSymbol=none]{H}{Q}{F}{M_8} \pstInterLC[PointSymbol=none,PointName=none]{Q}{M_1}{Q}{T}{M}{E} \pstInterLC[PointSymbol=none,PointName=none]{Q}{M_2}{Q}{T}{K}{C} \pstInterLC[PointSymbol=none,PointName=none]{Q}{M_3}{Q}{T}{I}{A} \pstInterLC[PointSymbol=none,PointName=none]{Q}{M_4}{Q}{T}{G}{O} \pswedge[fillstyle=solid,fillcolor=white](Q){8}{22.5}{45} \pswedge[fillstyle=solid,fillcolor=white](Q){8}{67.5}{90} \pswedge[fillstyle=solid,fillcolor=white](Q){8}{112.5}{135} \pswedge[fillstyle=solid,fillcolor=white](Q){8}{157.5}{180} \pswedge[fillstyle=solid,fillcolor=white](Q){8}{202.5}{225} \pswedge[fillstyle=solid,fillcolor=white](Q){8}{247.5}{270} \pswedge[fillstyle=solid,fillcolor=white](Q){8}{292.5}{315} \pswedge[fillstyle=solid,fillcolor=white](Q){8}{337.5}{360} \pspolygon(A)(B)(C)(D)(E)(F)(G)(H)(I)(J)(K)(L)(M)(N)(O)(P) \psline[fillstyle=solid,fillcolor=lightgray](B)(C)(Q) \psline[fillstyle=solid,fillcolor=lightgray](P)(A)(Q) \psline[fillstyle=solid,fillcolor=lightgray](N)(O)(Q) \psline[fillstyle=solid,fillcolor=lightgray](L)(M)(Q) \psline[fillstyle=solid,fillcolor=lightgray](J)(K)(Q) \psline[fillstyle=solid,fillcolor=lightgray](H)(I)(Q) \psline[fillstyle=solid,fillcolor=lightgray](F)(G)(Q) \psline[fillstyle=solid,fillcolor=lightgray](D)(E)(Q) \psline[fillstyle=solid,fillcolor=white](B)(A)(Q) \psline[fillstyle=solid,fillcolor=white](P)(O)(Q) \psline[fillstyle=solid,fillcolor=white](N)(M)(Q) \psline[fillstyle=solid,fillcolor=white](L)(K)(Q) \psline[fillstyle=solid,fillcolor=white](J)(I)(Q) \psline[fillstyle=solid,fillcolor=white](H)(G)(Q) \psline[fillstyle=solid,fillcolor=white](F)(E)(Q) \psline[fillstyle=solid,fillcolor=white](D)(C)(Q) \psline(N)(F) \psline(B)(J) \pstBissectBAC[PointSymbol=none,PointName=none]{F}{Q}{B}{D} \pstBissectBAC[PointSymbol=none,PointName=none]{B}{Q}{N}{P} \pstBissectBAC[PointSymbol=none,PointName=none]{N}{Q}{J}{L} \pstBissectBAC[PointSymbol=none,PointName=none]{J}{Q}{F}{H} \pstBissectBAC[PointName=none,PointSymbol=none]{F}{Q}{D}{M_1} \pstBissectBAC[PointName=none,PointSymbol=none]{D}{Q}{B}{M_2} \pstBissectBAC[PointName=none,PointSymbol=none]{B}{Q}{P}{M_3} \pstBissectBAC[PointName=none,PointSymbol=none]{P}{Q}{N}{M_4} \pstBissectBAC[PointName=none,PointSymbol=none]{N}{Q}{L}{M_5} \pstBissectBAC[PointName=none,PointSymbol=none]{L}{Q}{J}{M_6} \pstBissectBAC[PointName=none,PointSymbol=none]{J}{Q}{H}{M_7} \pstBissectBAC[PointName=none,PointSymbol=none]{H}{Q}{F}{M_8} \put(-1.5,-10){Figure 2.} \endpspicture \end{center} \end{multicols} \vskip 1.5cm \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 les longueurs données).}}\\ \vskip 0.5cm \shadowbox{ \begin{minipage}[c]{\textwidth} \begin{multicols}{2}\setlength{\columnseprule}{0.5pt} \textbf{\'Etape 1.}\\ Construire un cercle $\mathcal{C}$ de centre $Q$ et de rayon 8~cm.\\ \textbf{\'Etape 2.}\\ Construire tracer deux diamètres perpendiculaires $[NF]$ et $[BJ]$.\\ \textbf{\'Etape 3.}\\ Construire les bissectrices des quatre angles obtenus.\\ \textbf{\'Etape 4.}\\ Construire les bissectrices des huit nouveaux angles obtenus.\\ \textbf{\'Etape 5.}\\ Tracer le cercle de centre $Q$ et de rayon 4~cm.\\ \textbf{\'Etape 6.}\\ Nommer les points comme sur la figure 1.\\ \textbf{\'Etape 7.}\\ Tracer le polygone $ABCDEFGHIJKLMNOP$.\\ \textbf{\'Etape 8.}\\ Colorier la figure obtenue comme sur la figure 2 en alternant deux couleurs. \end{multicols} \end{minipage} } \end{document}