# index.tex

\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}}
\pagestyle{fancy}

\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

\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}