\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 : la spirale infernale !}}\\ \vskip 0.025cm \end{center}\\ \hline \end{tabularx} \end{center} \vskip 0.55cm Il s'agit de construire la figure ci-dessous. \begin{center} \psset{unit=0.75cm} \pspicture(-12,-12)(12,12) \psframe(-12,-12)(12,12) \pstGeonode[PointSymbol=none,PosAngle={180,0,90,-90,45},PointName=none](-12,0){x}(12,0){y}(0,12){u}(0,-12){v}(0,0){O} \pstLineAB{x}{y} \pstLineAB{u}{v} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-5,PointName=none]{y}{O}{u}{r} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-5,PointName=none]{u}{O}{x}{k} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-5,PointName=none]{x}{O}{v}{p} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-5,PointName=none]{v}{O}{y}{q} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{y}{O}{r}{t} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{r}{O}{u}{t'} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{u}{O}{k}{k'} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{k}{O}{x}{k''} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{x}{O}{p}{p'} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{p}{O}{v}{p''} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{v}{O}{q}{q'} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{q}{O}{y}{q''} \pstGeonode[PointSymbol=x,PosAngle={-90},PointName=none](12,0){A} \pstGeonode[PointSymbol=x,PosAngle={-90},PointName=none](0,12){B} \pstGeonode[PointSymbol=x,PosAngle={-90},PointName=none](-12,0){C} \pstGeonode[PointSymbol=x,PosAngle={-90},PointName=none](0,-12){D} % Projetés orthogonaux à partir de A \pstProjection[PointSymbol=x,PointName=none]{O}{t}{A}{A_1} \pstLineAB{A}{A_1} \pstProjection[PointSymbol=x,PointName=none]{O}{r}{A_1}{A_2} \pstLineAB{A_1}{A_2} \pstProjection[PointSymbol=x,PointName=none]{O}{t'}{A_2}{A_3} \pstLineAB{A_2}{A_3} \pstProjection[PointSymbol=x,PointName=none]{O}{u}{A_3}{A_4} \pstLineAB{A_3}{A_4} \pstProjection[PointSymbol=x,PointName=none]{O}{k'}{A_4}{A_5} \pstLineAB{A_4}{A_5} \pstProjection[PointSymbol=x,PointName=none]{O}{k}{A_5}{A_6} \pstLineAB{A_5}{A_6} \pstProjection[PointSymbol=x,PointName=none]{O}{k''}{A_6}{A_7} \pstLineAB{A_6}{A_7} \pstProjection[PointSymbol=x,PointName=none]{O}{x}{A_7}{A_8} \pstLineAB{A_7}{A_8} \pstProjection[PointSymbol=x,PointName=none]{O}{p'}{A_8}{A_9} \pstLineAB{A_8}{A_9} \pstProjection[PointSymbol=x,PointName=none]{O}{p}{A_9}{A_10} \pstLineAB{A_9}{A_10} \pstProjection[PointSymbol=x,PointName=none]{O}{p''}{A_10}{A_11} \pstLineAB{A_10}{A_11} \pstProjection[PointSymbol=x,PointName=none]{O}{v}{A_11}{A_12} \pstLineAB{A_11}{A_12} \pstProjection[PointSymbol=x,PointName=none]{O}{q'}{A_12}{A_13} \pstLineAB{A_12}{A_13} \pstProjection[PointSymbol=x,PointName=none]{O}{q}{A_13}{A_14} \pstLineAB{A_13}{A_14} \pstProjection[PointSymbol=x,PointName=none]{O}{q''}{A_14}{A_15} \pstLineAB{A_14}{A_15} \pstProjection[PointSymbol=x,PointName=none]{O}{y}{A_15}{A_16} \pstLineAB{A_15}{A_16} \pstProjection[PointSymbol=x,PointName=none]{O}{t}{A_16}{A_17} \pstLineAB{A_16}{A_17} \pstProjection[PointSymbol=x,PointName=none]{O}{r}{A_17}{A_18} \pstLineAB{A_17}{A_18} \pstProjection[PointSymbol=x,PointName=none]{O}{t'}{A_18}{A_19} \pstLineAB{A_18}{A_19} \pstProjection[PointSymbol=x,PointName=none]{O}{u}{A_19}{A_20} \pstLineAB{A_19}{A_20} \pstProjection[PointSymbol=x,PointName=none]{O}{k'}{A_20}{A_21} \pstLineAB{A_20}{A_21} \pstProjection[PointSymbol=x,PointName=none]{O}{k}{A_21}{A_22} \pstLineAB{A_21}{A_22} \pstProjection[PointSymbol=x,PointName=none]{O}{k''}{A_22}{A_23} \pstLineAB{A_22}{A_23} \pstProjection[PointSymbol=x,PointName=none]{O}{x}{A_23}{A_24} \pstLineAB{A_23}{A_24} \pstProjection[PointSymbol=x,PointName=none]{O}{p'}{A_24}{A_25} \pstLineAB{A_24}{A_25} \pstProjection[PointSymbol=x,PointName=none]{O}{p}{A_25}{A_26} \pstLineAB{A_25}{A_26} \pstProjection[PointSymbol=x,PointName=none]{O}{p''}{A_26}{A_27} \pstLineAB{A_26}{A_27} \pstProjection[PointSymbol=x,PointName=none]{O}{v}{A_27}{A_28} \pstLineAB{A_27}{A_28} \pstProjection[PointSymbol=x,PointName=none]{O}{q'}{A_28}{A_29} \pstLineAB{A_28}{A_29} \pstProjection[PointSymbol=x,PointName=none]{O}{q}{A_29}{A_30} \pstLineAB{A_29}{A_30} \pstProjection[PointSymbol=x,PointName=none]{O}{q''}{A_30}{A_31} \pstLineAB{A_30}{A_31} \pstProjection[PointSymbol=x,PointName=none]{O}{y}{A_31}{A_32} \pstLineAB{A_31}{A_32} \pstProjection[PointSymbol=x,PointName=none]{O}{t}{A_32}{A_33} \pstLineAB{A_32}{A_33} \pstProjection[PointSymbol=x,PointName=none]{O}{r}{A_33}{A_34} \pstLineAB{A_33}{A_34} \pstProjection[PointSymbol=x,PointName=none]{O}{t'}{A_34}{A_35} \pstLineAB{A_34}{A_35} \pstProjection[PointSymbol=x,PointName=none]{O}{u}{A_35}{A_36} \pstLineAB{A_35}{A_36} \pstProjection[PointSymbol=x,PointName=none]{O}{k'}{A_36}{A_37} \pstLineAB{A_36}{A_37} \pstProjection[PointSymbol=x,PointName=none]{O}{k}{A_37}{A_38} \pstLineAB{A_37}{A_38} \pstProjection[PointSymbol=x,PointName=none]{O}{k''}{A_38}{A_39} \pstLineAB{A_38}{A_39} \pstProjection[PointSymbol=x,PointName=none]{O}{x}{A_39}{A_40} \pstLineAB{A_39}{A_40} \pstProjection[PointSymbol=x,PointName=none]{O}{p'}{A_40}{A_41} \pstLineAB{A_40}{A_41} \pstProjection[PointSymbol=x,PointName=none]{O}{p}{A_41}{A_42} \pstLineAB{A_41}{A_42} \pstProjection[PointSymbol=x,PointName=none]{O}{p''}{A_42}{A_43} \pstLineAB{A_42}{A_43} \pstProjection[PointSymbol=x,PointName=none]{O}{v}{A_43}{A_44} \pstLineAB{A_43}{A_44} \pstProjection[PointSymbol=x,PointName=none]{O}{q'}{A_44}{A_45} \pstLineAB{A_44}{A_45} % Projetés orthogonaux à partir de B \pstProjection[PointSymbol=x,PointName=none]{O}{k'}{B}{B_1} \pstLineAB{B}{B_1} \pstProjection[PointSymbol=x,PointName=none]{O}{k}{B_1}{B_2} \pstLineAB{B_1}{B_2} \pstProjection[PointSymbol=x,PointName=none]{O}{k''}{B_2}{B_3} \pstLineAB{B_2}{B_3} \pstProjection[PointSymbol=x,PointName=none]{O}{x}{B_3}{B_4} \pstLineAB{B_3}{B_4} \pstProjection[PointSymbol=x,PointName=none]{O}{p'}{B_4}{B_5} \pstLineAB{B_4}{B_5} \pstProjection[PointSymbol=x,PointName=none]{O}{p}{B_5}{B_6} \pstLineAB{B_5}{B_6} \pstProjection[PointSymbol=x,PointName=none]{O}{p''}{B_6}{B_7} \pstLineAB{B_6}{B_7} \pstProjection[PointSymbol=x,PointName=none]{O}{v}{B_7}{B_8} \pstLineAB{B_7}{B_8} \pstProjection[PointSymbol=x,PointName=none]{O}{q'}{B_8}{B_9} \pstLineAB{B_8}{B_9} \pstProjection[PointSymbol=x,PointName=none]{O}{q}{B_9}{B_10} \pstLineAB{B_9}{B_10} \pstProjection[PointSymbol=x,PointName=none]{O}{q''}{B_10}{B_11} \pstLineAB{B_10}{B_11} \pstProjection[PointSymbol=x,PointName=none]{O}{y}{B_11}{B_12} \pstLineAB{B_11}{B_12} \pstProjection[PointSymbol=x,PointName=none]{O}{t}{B_12}{B_13} \pstLineAB{B_12}{B_13} \pstProjection[PointSymbol=x,PointName=none]{O}{r}{B_13}{B_14} \pstLineAB{B_13}{B_14} \pstProjection[PointSymbol=x,PointName=none]{O}{t'}{B_14}{B_15} \pstLineAB{B_14}{B_15} \pstProjection[PointSymbol=x,PointName=none]{O}{u}{B_15}{B_16} \pstLineAB{B_15}{B_16} \pstProjection[PointSymbol=x,PointName=none]{O}{k'}{B_16}{B_17} \pstLineAB{B_16}{B_17} \pstProjection[PointSymbol=x,PointName=none]{O}{k}{B_17}{B_18} \pstLineAB{B_17}{B_18} \pstProjection[PointSymbol=x,PointName=none]{O}{k''}{B_18}{B_19} \pstLineAB{B_18}{B_19} \pstProjection[PointSymbol=x,PointName=none]{O}{x}{B_19}{B_20} \pstLineAB{B_19}{B_20} \pstProjection[PointSymbol=x,PointName=none]{O}{p'}{B_20}{B_21} \pstLineAB{B_20}{B_21} \pstProjection[PointSymbol=x,PointName=none]{O}{p}{B_21}{B_22} \pstLineAB{B_21}{B_22} \pstProjection[PointSymbol=x,PointName=none]{O}{p''}{B_22}{B_23} \pstLineAB{B_22}{B_23} \pstProjection[PointSymbol=x,PointName=none]{O}{v}{B_23}{B_24} \pstLineAB{B_23}{B_24} \pstProjection[PointSymbol=x,PointName=none]{O}{q'}{B_24}{B_25} \pstLineAB{B_24}{B_25} \pstProjection[PointSymbol=x,PointName=none]{O}{q}{B_25}{B_26} \pstLineAB{B_25}{B_26} \pstProjection[PointSymbol=x,PointName=none]{O}{q''}{B_26}{B_27} \pstLineAB{B_26}{B_27} \pstProjection[PointSymbol=x,PointName=none]{O}{y}{B_27}{B_28} \pstLineAB{B_27}{B_28} \pstProjection[PointSymbol=x,PointName=none]{O}{t}{B_28}{B_29} \pstLineAB{B_28}{B_29} \pstProjection[PointSymbol=x,PointName=none]{O}{r}{B_29}{B_30} \pstLineAB{B_29}{B_30} \pstProjection[PointSymbol=x,PointName=none]{O}{t'}{B_30}{B_31} \pstLineAB{B_30}{B_31} \pstProjection[PointSymbol=x,PointName=none]{O}{u}{B_31}{B_32} \pstLineAB{B_31}{B_32} \pstProjection[PointSymbol=x,PointName=none]{O}{k'}{B_32}{B_33} \pstLineAB{B_32}{B_33} \pstProjection[PointSymbol=x,PointName=none]{O}{k}{B_33}{B_34} \pstLineAB{B_33}{B_34} \pstProjection[PointSymbol=x,PointName=none]{O}{k''}{B_34}{B_35} \pstLineAB{B_34}{B_35} \pstProjection[PointSymbol=x,PointName=none]{O}{x}{B_35}{B_36} \pstLineAB{B_35}{B_36} \pstProjection[PointSymbol=x,PointName=none]{O}{p'}{B_36}{B_37} \pstLineAB{B_36}{B_37} \pstProjection[PointSymbol=x,PointName=none]{O}{p}{B_37}{B_38} \pstLineAB{B_37}{B_38} \pstProjection[PointSymbol=x,PointName=none]{O}{p''}{B_38}{B_39} \pstLineAB{B_38}{B_39} \pstProjection[PointSymbol=x,PointName=none]{O}{v}{B_39}{B_40} \pstLineAB{B_39}{B_40} \pstProjection[PointSymbol=x,PointName=none]{O}{q'}{B_40}{B_41} \pstLineAB{B_40}{B_41} \pstProjection[PointSymbol=x,PointName=none]{O}{q}{B_41}{B_42} \pstLineAB{B_41}{B_42} \pstProjection[PointSymbol=x,PointName=none]{O}{q''}{B_42}{B_43} \pstLineAB{B_42}{B_43} \pstProjection[PointSymbol=x,PointName=none]{O}{y}{B_43}{B_44} \pstLineAB{B_43}{B_44} \pstProjection[PointSymbol=x,PointName=none]{O}{t}{B_44}{B_45} \pstLineAB{B_44}{B_45} % Projetés orthogonaux à partir de C \pstProjection[PointSymbol=x=none,PointName=none]{O}{p'}{C}{C_1} \pstLineAB{C}{C_1} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p}{C_1}{C_2} \pstLineAB{C_1}{C_2} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p''}{C_2}{C_3} \pstLineAB{C_2}{C_3} \pstProjection[PointSymbol=x=none,PointName=none]{O}{v}{C_3}{C_4} \pstLineAB{C_3}{C_4} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q'}{C_4}{C_5} \pstLineAB{C_4}{C_5} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q}{C_5}{C_6} \pstLineAB{C_5}{C_6} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q''}{C_6}{C_7} \pstLineAB{C_6}{C_7} \pstProjection[PointSymbol=x=none,PointName=none]{O}{y}{C_7}{C_8} \pstLineAB{C_7}{C_8} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t}{C_8}{C_9} \pstLineAB{C_8}{C_9} \pstProjection[PointSymbol=x=none,PointName=none]{O}{r}{C_9}{C_10} \pstLineAB{C_9}{C_10} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t'}{C_10}{C_11} \pstLineAB{C_10}{C_11} \pstProjection[PointSymbol=x=none,PointName=none]{O}{u}{C_11}{C_12} \pstLineAB{C_11}{C_12} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k'}{C_12}{C_13} \pstLineAB{C_12}{C_13} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k}{C_13}{C_14} \pstLineAB{C_13}{C_14} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k''}{C_14}{C_15} \pstLineAB{C_14}{C_15} \pstProjection[PointSymbol=x=none,PointName=none]{O}{x}{C_15}{C_16} \pstLineAB{C_15}{C_16} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p'}{C_16}{C_17} \pstLineAB{C_16}{C_17} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p}{C_17}{C_18} \pstLineAB{C_17}{C_18} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p''}{C_18}{C_19} \pstLineAB{C_18}{C_19} \pstProjection[PointSymbol=x=none,PointName=none]{O}{v}{C_19}{C_20} \pstLineAB{C_19}{C_20} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q'}{C_20}{C_21} \pstLineAB{C_20}{C_21} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q}{C_21}{C_22} \pstLineAB{C_21}{C_22} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q''}{C_22}{C_23} \pstLineAB{C_22}{C_23} \pstProjection[PointSymbol=x=none,PointName=none]{O}{y}{C_23}{C_24} \pstLineAB{C_23}{C_24} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t}{C_24}{C_25} \pstLineAB{C_24}{C_25} \pstProjection[PointSymbol=x=none,PointName=none]{O}{r}{C_25}{C_26} \pstLineAB{C_25}{C_26} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t'}{C_26}{C_27} \pstLineAB{C_26}{C_27} \pstProjection[PointSymbol=x=none,PointName=none]{O}{u}{C_27}{C_28} \pstLineAB{C_27}{C_28} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k'}{C_28}{C_29} \pstLineAB{C_28}{C_29} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k}{C_29}{C_30} \pstLineAB{C_29}{C_30} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k''}{C_30}{C_31} \pstLineAB{C_30}{C_31} \pstProjection[PointSymbol=x=none,PointName=none]{O}{x}{C_31}{C_32} \pstLineAB{C_31}{C_32} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p'}{C_32}{C_33} \pstLineAB{C_32}{C_33} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p}{C_33}{C_34} \pstLineAB{C_33}{C_34} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p''}{C_34}{C_35} \pstLineAB{C_34}{C_35} \pstProjection[PointSymbol=x=none,PointName=none]{O}{v}{C_35}{C_36} \pstLineAB{C_35}{C_36} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q'}{C_36}{C_37} \pstLineAB{C_36}{C_37} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q}{C_37}{C_38} \pstLineAB{C_37}{C_38} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q''}{C_38}{C_39} \pstLineAB{C_38}{C_39} \pstProjection[PointSymbol=x=none,PointName=none]{O}{y}{C_39}{C_40} \pstLineAB{C_39}{C_40} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t}{C_40}{C_41} \pstLineAB{C_40}{C_41} \pstProjection[PointSymbol=x=none,PointName=none]{O}{r}{C_41}{C_42} \pstLineAB{C_41}{C_42} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t'}{C_42}{C_43} \pstLineAB{C_42}{C_43} \pstProjection[PointSymbol=x=none,PointName=none]{O}{u}{C_43}{C_44} \pstLineAB{C_43}{C_44} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k'}{C_44}{C_45} \pstLineAB{C_44}{C_45} % Projetés orthogonaux à partir de D \pstProjection[PointSymbol=x=none,PointName=none]{O}{q'}{D}{D_1} \pstLineAB{D}{D_1} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q}{D_1}{D_2} \pstLineAB{D_1}{D_2} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q''}{D_2}{D_3} \pstLineAB{D_2}{D_3} \pstProjection[PointSymbol=x=none,PointName=none]{O}{y}{D_3}{D_4} \pstLineAB{D_3}{D_4} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t}{D_4}{D_5} \pstLineAB{D_4}{D_5} \pstProjection[PointSymbol=x=none,PointName=none]{O}{r}{D_5}{D_6} \pstLineAB{D_5}{D_6} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t'}{D_6}{D_7} \pstLineAB{D_6}{D_7} \pstProjection[PointSymbol=x=none,PointName=none]{O}{u}{D_7}{D_8} \pstLineAB{D_7}{D_8} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k'}{D_8}{D_9} \pstLineAB{D_8}{D_9} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k}{D_9}{D_10} \pstLineAB{D_9}{D_10} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k''}{D_10}{D_11} \pstLineAB{D_10}{D_11} \pstProjection[PointSymbol=x=none,PointName=none]{O}{x}{D_11}{D_12} \pstLineAB{D_11}{D_12} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p'}{D_12}{D_13} \pstLineAB{D_12}{D_13} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p}{D_13}{D_14} \pstLineAB{D_13}{D_14} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p''}{D_14}{D_15} \pstLineAB{D_14}{D_15} \pstProjection[PointSymbol=x=none,PointName=none]{O}{v}{D_15}{D_16} \pstLineAB{D_15}{D_16} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q'}{D_16}{D_17} \pstLineAB{D_16}{D_17} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q}{D_17}{D_18} \pstLineAB{D_17}{D_18} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q''}{D_18}{D_19} \pstLineAB{D_18}{D_19} \pstProjection[PointSymbol=x=none,PointName=none]{O}{y}{D_19}{D_20} \pstLineAB{D_19}{D_20} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t}{D_20}{D_21} \pstLineAB{D_20}{D_21} \pstProjection[PointSymbol=x=none,PointName=none]{O}{r}{D_21}{D_22} \pstLineAB{D_21}{D_22} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t'}{D_22}{D_23} \pstLineAB{D_22}{D_23} \pstProjection[PointSymbol=x=none,PointName=none]{O}{u}{D_23}{D_24} \pstLineAB{D_23}{D_24} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k'}{D_24}{D_25} \pstLineAB{D_24}{D_25} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k}{D_25}{D_26} \pstLineAB{D_25}{D_26} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k''}{D_26}{D_27} \pstLineAB{D_26}{D_27} \pstProjection[PointSymbol=x=none,PointName=none]{O}{x}{D_27}{D_28} \pstLineAB{D_27}{D_28} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p'}{D_28}{D_29} \pstLineAB{D_28}{D_29} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p}{D_29}{D_30} \pstLineAB{D_29}{D_30} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p''}{D_30}{D_31} \pstLineAB{D_30}{D_31} \pstProjection[PointSymbol=x=none,PointName=none]{O}{v}{D_31}{D_32} \pstLineAB{D_31}{D_32} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q'}{D_32}{D_33} \pstLineAB{D_32}{D_33} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q}{D_33}{D_34} \pstLineAB{D_33}{D_34} \pstProjection[PointSymbol=x=none,PointName=none]{O}{q''}{D_34}{D_35} \pstLineAB{D_34}{D_35} \pstProjection[PointSymbol=x=none,PointName=none]{O}{y}{D_35}{D_36} \pstLineAB{D_35}{D_36} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t}{D_36}{D_37} \pstLineAB{D_36}{D_37} \pstProjection[PointSymbol=x=none,PointName=none]{O}{r}{D_37}{D_38} \pstLineAB{D_37}{D_38} \pstProjection[PointSymbol=x=none,PointName=none]{O}{t'}{D_38}{D_39} \pstLineAB{D_38}{D_39} \pstProjection[PointSymbol=x=none,PointName=none]{O}{u}{D_39}{D_40} \pstLineAB{D_39}{D_40} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k'}{D_40}{D_41} \pstLineAB{D_40}{D_41} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k}{D_41}{D_42} \pstLineAB{D_41}{D_42} \pstProjection[PointSymbol=x=none,PointName=none]{O}{k''}{D_42}{D_43} \pstLineAB{D_42}{D_43} \pstProjection[PointSymbol=x=none,PointName=none]{O}{x}{D_43}{D_44} \pstLineAB{D_43}{D_44} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p'}{D_44}{D_45} \pstLineAB{D_44}{D_45} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p}{D_45}{D_46} \pstLineAB{D_45}{D_46} \pstProjection[PointSymbol=x=none,PointName=none]{O}{p''}{D_46}{D_47} \pstLineAB{D_46}{D_47} \pstProjection[PointSymbol=x=none,PointName=none]{O}{v}{D_47}{D_48} \pstLineAB{D_47}{D_48} % Nouveaux points \pstGeonode[PointSymbol=none,PointName=none](12,12){T}(-12,12){U}(-12,-12){V}(12,-12){W} %,PointName=none \pstInterLL[PointName=none,PointSymbol=none]{T}{U}{O}{t'}{a_1} \pstInterLL[PointName=none,PointSymbol=none]{T}{U}{O}{k'}{a_2} \pstInterLL[PointName=none,PointSymbol=none]{U}{V}{O}{k''}{a_3} \pstInterLL[PointName=none,PointSymbol=none]{U}{V}{O}{p'}{a_4} \pstInterLL[PointName=none,PointSymbol=none]{W}{V}{O}{p''}{a_5} \pstInterLL[PointName=none,PointSymbol=none]{W}{V}{O}{q'}{a_6} \pstInterLL[PointName=none,PointSymbol=none]{W}{T}{O}{q''}{a_7} \pstInterLL[PointName=none,PointSymbol=none]{W}{T}{O}{t}{a_8} % polygones pleins \pspolygon[fillstyle=solid,fillcolor=black](A_1)(A_2)(D_6)(D_5) \pspolygon[fillstyle=solid,fillcolor=black](A_3)(A_4)(D_8)(D_7) \pspolygon[fillstyle=solid,fillcolor=black](A_5)(A_6)(D_10)(D_9) \pspolygon[fillstyle=solid,fillcolor=black](A_7)(A_8)(D_12)(D_11) \pspolygon[fillstyle=solid,fillcolor=black](A_9)(A_10)(D_14)(D_13) \pspolygon[fillstyle=solid,fillcolor=black](A_11)(A_12)(D_16)(D_15) \pspolygon[fillstyle=solid,fillcolor=black](A_13)(A_14)(D_18)(D_17) \pspolygon[fillstyle=solid,fillcolor=black](A_15)(A_16)(D_20)(D_19) \pspolygon[fillstyle=solid,fillcolor=black](A_17)(A_18)(D_22)(D_21) \pspolygon[fillstyle=solid,fillcolor=black](A_19)(A_20)(D_24)(D_23) \pspolygon[fillstyle=solid,fillcolor=black](A_21)(A_22)(D_26)(D_25) \pspolygon[fillstyle=solid,fillcolor=black](A_23)(A_24)(D_28)(D_27) \pspolygon[fillstyle=solid,fillcolor=black](A_25)(A_26)(D_30)(D_29) \pspolygon[fillstyle=solid,fillcolor=black](A_27)(A_28)(D_32)(D_31) \pspolygon[fillstyle=solid,fillcolor=black](A_29)(A_30)(D_34)(D_33) \pspolygon[fillstyle=solid,fillcolor=black](A_31)(A_32)(D_36)(D_35) \pspolygon[fillstyle=solid,fillcolor=black](A_33)(A_34)(D_38)(D_37) \pspolygon[fillstyle=solid,fillcolor=black](A_35)(A_36)(D_40)(D_39) \pspolygon[fillstyle=solid,fillcolor=black](B_2)(B_3)(A_7)(A_6) \pspolygon[fillstyle=solid,fillcolor=black](B_4)(B_5)(A_9)(A_8) \pspolygon[fillstyle=solid,fillcolor=black](B_6)(B_7)(A_11)(A_10) \pspolygon[fillstyle=solid,fillcolor=black](B_8)(B_9)(A_13)(A_12) \pspolygon[fillstyle=solid,fillcolor=black](B_10)(B_11)(A_15)(A_14) \pspolygon[fillstyle=solid,fillcolor=black](B_12)(B_13)(A_17)(A_16) \pspolygon[fillstyle=solid,fillcolor=black](B_14)(B_15)(A_19)(A_18) \pspolygon[fillstyle=solid,fillcolor=black](B_16)(B_17)(A_21)(A_20) \pspolygon[fillstyle=solid,fillcolor=black](B_18)(B_19)(A_23)(A_22) \pspolygon[fillstyle=solid,fillcolor=black](B_20)(B_21)(A_25)(A_24) \pspolygon[fillstyle=solid,fillcolor=black](B_22)(B_23)(A_27)(A_26) \pspolygon[fillstyle=solid,fillcolor=black](B_24)(B_25)(A_29)(A_28) \pspolygon[fillstyle=solid,fillcolor=black](B_26)(B_27)(A_31)(A_30) \pspolygon[fillstyle=solid,fillcolor=black](B_28)(B_29)(A_33)(A_32) \pspolygon[fillstyle=solid,fillcolor=black](B_30)(B_31)(A_35)(A_34) \pspolygon[fillstyle=solid,fillcolor=black](B_32)(B_33)(A_37)(A_36) \pspolygon[fillstyle=solid,fillcolor=black](B_34)(B_35)(A_39)(A_38) \pspolygon[fillstyle=solid,fillcolor=black](B_36)(B_37)(A_41)(A_40) \pspolygon[fillstyle=solid,fillcolor=black](B)(B_1)(A_5)(A_4) \pspolygon[fillstyle=solid,fillcolor=black](T)(a_1)(A_3)(A_2) \pspolygon[fillstyle=solid,fillcolor=black](A_1)(a_8)(A) \pspolygon[fillstyle=solid,fillcolor=black](C_1)(C_2)(B_6)(B_5) \pspolygon[fillstyle=solid,fillcolor=black](C_3)(C_4)(B_8)(B_7) \pspolygon[fillstyle=solid,fillcolor=black](C_5)(C_6)(B_10)(B_9) \pspolygon[fillstyle=solid,fillcolor=black](C_7)(C_8)(B_12)(B_11) \pspolygon[fillstyle=solid,fillcolor=black](C_9)(C_10)(B_14)(B_13) \pspolygon[fillstyle=solid,fillcolor=black](C_11)(C_12)(B_16)(B_15) \pspolygon[fillstyle=solid,fillcolor=black](C_13)(C_14)(B_18)(B_17) \pspolygon[fillstyle=solid,fillcolor=black](C_15)(C_16)(B_20)(B_19) \pspolygon[fillstyle=solid,fillcolor=black](C_17)(C_18)(B_22)(B_21) \pspolygon[fillstyle=solid,fillcolor=black](C_19)(C_20)(B_24)(B_23) \pspolygon[fillstyle=solid,fillcolor=black](C_21)(C_22)(B_26)(B_25) \pspolygon[fillstyle=solid,fillcolor=black](C_23)(C_24)(B_28)(B_27) \pspolygon[fillstyle=solid,fillcolor=black](C_25)(C_26)(B_30)(B_29) \pspolygon[fillstyle=solid,fillcolor=black](C_27)(C_28)(B_32)(B_31) \pspolygon[fillstyle=solid,fillcolor=black](C_29)(C_30)(B_34)(B_33) \pspolygon[fillstyle=solid,fillcolor=black](C_31)(C_32)(B_36)(B_35) \pspolygon[fillstyle=solid,fillcolor=black](C_33)(C_34)(B_38)(B_37) \pspolygon[fillstyle=solid,fillcolor=black](C_35)(C_36)(B_40)(B_39) \pspolygon[fillstyle=solid,fillcolor=black](a_2)(U)(B_2)(B_1) \pspolygon[fillstyle=solid,fillcolor=black](B_3)(a_3)(C)(B_4) \pspolygon[fillstyle=solid,fillcolor=black](D_2)(D_3)(C_7)(C_6) \pspolygon[fillstyle=solid,fillcolor=black](D_4)(D_5)(C_9)(C_8) \pspolygon[fillstyle=solid,fillcolor=black](D_6)(D_7)(C_11)(C_10) \pspolygon[fillstyle=solid,fillcolor=black](D_8)(D_9)(C_13)(C_12) \pspolygon[fillstyle=solid,fillcolor=black](D_10)(D_11)(C_15)(C_14) \pspolygon[fillstyle=solid,fillcolor=black](D_12)(D_13)(C_17)(C_16) \pspolygon[fillstyle=solid,fillcolor=black](D_14)(D_15)(C_19)(C_18) \pspolygon[fillstyle=solid,fillcolor=black](D_16)(D_17)(C_21)(C_20) \pspolygon[fillstyle=solid,fillcolor=black](D_18)(D_19)(C_23)(C_22) \pspolygon[fillstyle=solid,fillcolor=black](D_20)(D_21)(C_25)(C_24) \pspolygon[fillstyle=solid,fillcolor=black](D_22)(D_23)(C_27)(C_26) \pspolygon[fillstyle=solid,fillcolor=black](D_24)(D_25)(C_29)(C_28) \pspolygon[fillstyle=solid,fillcolor=black](D_26)(D_27)(C_31)(C_30) \pspolygon[fillstyle=solid,fillcolor=black](D_28)(D_29)(C_33)(C_32) \pspolygon[fillstyle=solid,fillcolor=black](D_30)(D_31)(C_35)(C_34) \pspolygon[fillstyle=solid,fillcolor=black](D_32)(D_33)(C_37)(C_36) \pspolygon[fillstyle=solid,fillcolor=black](D_34)(D_35)(C_39)(C_38) \pspolygon[fillstyle=solid,fillcolor=black](D_36)(D_37)(C_41)(C_40) \pspolygon[fillstyle=solid,fillcolor=black](D)(D_1)(C_5)(C_4) \pspolygon[fillstyle=solid,fillcolor=black](a_5)(C_3)(C_2)(V) \pspolygon[fillstyle=solid,fillcolor=black](C)(a_4)(C_1) \pspolygon[fillstyle=solid,fillcolor=black](D_1)(D_2)(W)(a_6) \pspolygon[fillstyle=solid,fillcolor=black](A)(D_4)(D_3)(a_7) \endpspicture \end{center} \newpage Une figure intermédiaire est la figure suivante. \begin{center} \psset{unit=0.55cm} \pspicture(-12,-12)(12,12) \psframe(-12,-12)(12,12) \pstGeonode[PointSymbol=none,PosAngle={135,45,45,-45,45}](-12,0){x}(12,0){y}(0,12){u}(0,-12){v}(0,0){O} \pstLineAB{x}{y} \pstLineAB{u}{v} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-5]{y}{O}{u}{r} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-5,PointName=none]{u}{O}{x}{k} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-5,PointName=none]{x}{O}{v}{p} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-5,PointName=none]{v}{O}{y}{q} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1]{y}{O}{r}{t} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{r}{O}{u}{t'} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{u}{O}{k}{k'} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{k}{O}{x}{k''} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{x}{O}{p}{p'} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{p}{O}{v}{p''} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{v}{O}{q}{q'} \pstBissectBAC[PointSymbol=none,PosAngle=135,nodesep=-1,PointName=none]{q}{O}{y}{q''} \pstGeonode[PointSymbol=x,PosAngle={0}](12,0){A} \pstGeonode[PointSymbol=x,PosAngle={90}](0,12){B} \pstGeonode[PointSymbol=x,PosAngle={180}](-12,0){C} \pstGeonode[PointSymbol=x,PosAngle={-90}](0,-12){D} % Projetés orthogonaux à partir de A \pstProjection[PointSymbol=x,]{O}{t}{A}{A_1} \pstLineAB{A}{A_1} \pstProjection[PointSymbol=x]{O}{r}{A_1}{A_2} \pstLineAB{A_1}{A_2} \pstProjection[PointSymbol=x]{O}{t'}{A_2}{A_3} \pstLineAB{A_2}{A_3} \pstProjection[PointSymbol=x]{O}{u}{A_3}{A_4} \pstLineAB{A_3}{A_4} \pstRightAngle[RightAngleSize=0.5]{A}{A_1}{O} \pstRightAngle[RightAngleSize=0.5]{A_1}{A_2}{O} \pstRightAngle[RightAngleSize=0.5]{A_2}{A_3}{O} \pstRightAngle[RightAngleSize=0.5]{A_3}{A_4}{O} % Projetés orthogonaux à partir de B \pstProjection[PointSymbol=x]{O}{k'}{B}{B_1} \pstLineAB{B}{B_1} \pstProjection[PointSymbol=x]{O}{k}{B_1}{B_2} \pstLineAB{B_1}{B_2} \pstProjection[PointSymbol=x]{O}{k''}{B_2}{B_3} \pstLineAB{B_2}{B_3} \pstProjection[PointSymbol=x]{O}{x}{B_3}{B_4} \pstLineAB{B_3}{B_4} \pstRightAngle[RightAngleSize=0.5]{B}{B_1}{O} \pstRightAngle[RightAngleSize=0.5]{B_1}{B_2}{O} \pstRightAngle[RightAngleSize=0.5]{B_2}{B_3}{O} \pstRightAngle[RightAngleSize=0.5]{B_3}{B_4}{O} % Projetés orthogonaux à partir de C \pstProjection[PointSymbol=x]{O}{p'}{C}{C_1} \pstLineAB{C}{C_1} \pstProjection[PointSymbol=x]{O}{p}{C_1}{C_2} \pstLineAB{C_1}{C_2} \pstProjection[PointSymbol=x]{O}{p''}{C_2}{C_3} \pstLineAB{C_2}{C_3} \pstProjection[PointSymbol=x]{O}{v}{C_3}{C_4} \pstLineAB{C_3}{C_4} \pstRightAngle[RightAngleSize=0.5]{C}{C_1}{O} \pstRightAngle[RightAngleSize=0.5]{C_1}{C_2}{O} \pstRightAngle[RightAngleSize=0.5]{C_2}{C_3}{O} \pstRightAngle[RightAngleSize=0.5]{C_3}{C_4}{O} % Projetés orthogonaux à partir de D \pstProjection[PointSymbol=x]{O}{q'}{D}{D_1} \pstLineAB{D}{D_1} \pstProjection[PointSymbol=x]{O}{q}{D_1}{D_2} \pstLineAB{D_1}{D_2} \pstProjection[PointSymbol=x]{O}{q''}{D_2}{D_3} \pstLineAB{D_2}{D_3} \pstProjection[PointSymbol=x]{O}{y}{D_3}{D_4} \pstLineAB{D_3}{D_4} \pstRightAngle[RightAngleSize=0.5]{D}{D_1}{O} \pstRightAngle[RightAngleSize=0.5]{D_1}{D_2}{O} \pstRightAngle[RightAngleSize=0.5]{D_2}{D_3}{O} \pstRightAngle[RightAngleSize=0.5]{D_3}{D_4}{O} \endpspicture \end{center} \vskip 1.5cm \textbf{\LARGE Programme de construction }\\ \large{\texttt{Toutes les constructions doivent se faire à l'équerre, à la règle et au compas.}}\\ \vskip 0.5cm \shadowbox{ \begin{minipage}[c]{\textwidth} \begin{multicols}{2}\setlength{\columnseprule}{0.5pt} \textbf{\'Etape 1.}\\ Tracer deux droites perpendiculaires $(xy)$ et $(uv)$ qui se coupent en $O$.\\ \textbf{\'Etape 2.}\\ Tracer les bissectrices des quatre angles obtenus, puis les bissectrices des huit angles obtenus.\\ \textbf{\'Etape 3.}\\ Sur la demi-droite $[Oy)$, placer le point $A$ à 9~cm du point $O$, puis tracer la perpendiculaire abaissée de $A$ sur $[Ot)$. On obtient le point $A_1$. Tracer la perpendiculaire abaissée de $A_1$ sur $[Or)$. On obtient le point $A_2$. Continuer ainsi en tournant toujours dans le même sens.\\ \textbf{\'Etape 4.}\\ placer le point $B$ sur $[Ou)$ à 9~cm du point $O$, puis faire à partir de ce point les mêmes tracés perpendiculaires.\\ \textbf{\'Etape 5.}\\ Recommencer encore à partir des points $C$ et $D$ sur $[Ox)$ et $[Oy)$. Tourner toujours dans le même sens.\\ \textbf{\'Etape 6.}\\ Tracer ensuite le carré dont les axes de symétrie sont $(xy)$ et $(uv)$.\\ \textbf{\'Etape 7.}\\ Colorier de deux couleurs différentes afin d'obtenir la figure voulue.\\ \end{multicols} \end{minipage} } \end{document}