\documentclass[12pt]{article}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[francais]{babel}
\usepackage{lmodern}
\usepackage[colorlinks=true,urlcolor=blue!50!black, linkcolor=blue!50!black]{hyperref}
\usepackage{pas-cours}
\usepackage[tikz]{bclogo}
\usepackage{mathrsfs}
\usepackage{tkz-euclide}
\usepackage{pas-math}
\usepackage[vmargin=1.5cm,rmargin=2cm,lmargin=1.5cm]{geometry}
\begin{document}
\chap[num=A,color=blue]{Construction d'une astroïde}{Stéphane PASQUET, \today, \url{http://www.mathweb.fr}}

\itemclass{blue}
\begin{bclogo}[%
	couleur=blue!20,%
	couleurBord=blue!50!black,%
	arrondi=0.1,%
	logo=\bcquestion,%
	ombre=true,%
	couleurOmbre=blue!50!black!50,%
	blur,%
	barre=snake]{Intér\^ets de l'activité }
\begin{enumerate}
\item Tracer des perpendiculaires
\item Tracer des angles
\item Programme de construction
\item Symétrie axiale
\end{enumerate}
\end{bclogo}

\bigskip

\begin{pasbox}[style=act]
\itemclass{red}
\begin{enumerate}
\item  Construire un cercle $\mathscr{C}$ de centre O et de rayon 4 cm.
\item Construire deux diamètres perpendiculaires [AB] et [XY] de $\mathscr{C}$.
\item Soit un point M sur $\mathscr{C}$ tel que $\angle{BOM}=20^\circ$.
\begin{enumerate}
\item Construire en trait léger la droite perpendiculaire à [AB] passant par M ; elle coupe [AB] en D.
\item Construire en trait léger la droite perpendiculaire à [XY] passant par M ; elle coupe [XY] en E.
\end{enumerate}
\item Tracer en trait léger [DE].
\item Tracer en trait léger la droite perpendiculaire à [DE] passant par M ; elle coupe [DE] en P. Marquer uniquement le point P en rouge sans indiquer son nom.
\item Prendre d'autres points M sur $\mathscr{C}$ en allant par exemple de $10^\circ$ en $10^\circ$ jusqu'à $90^\circ$ puis construire pour chacun d'eux le point P. En prenant le symétriques des points obtenus par rapport à (AB) et (XY), puis en les reliant, on obtient une astroïde.
\end{enumerate}
\end{pasbox}

\newpage

\begin{enumerate}
\item \par~
\begin{center}
\begin{tikzpicture}
\tkzDefPoint(0,0){O}
\tkzDrawPoints(O)
\tkzLabelPoints[above](O)
\tkzDrawCircle[R](O,4cm)
\node at (3,4) {$\mathscr{C}$};
\end{tikzpicture}
\end{center}

\item\par~
\begin{center}
\begin{tikzpicture}
\tkzDefPoint(0,0){O}
\tkzDefPoint(-4,0){A}
\tkzDefPoint(4,0){B}
\tkzDefPoint(0,-4){X}
\tkzDefPoint(0,4){Y}
\tkzDrawPoints(O)
\tkzLabelPoints[above](Y)
\tkzLabelPoints[below right](O)
\tkzLabelPoints[below](X)
\tkzLabelPoints[left](A)
\tkzLabelPoints[right](B)
\tkzDrawCircle[R](O,4cm)
\node at (3,4) {$\mathscr{C}$};
\tkzDrawSegments(A,B X,Y)
\end{tikzpicture}
\end{center}

\newpage

\item
\begin{enumerate}
\item\par~
\begin{center}
\begin{tikzpicture}
\clip (-4.5,-4.5) rectangle (4.5,4.5);
\tkzDefPoint(0,0){O}
\tkzDefPoint(-4,0){A}
\tkzDefPoint(4,0){B}
\tkzDefPoint(0,-4){X}
\tkzDefPoint(0,4){Y}
\tkzDrawPoints(O)
\tkzLabelPoints[above](Y)
\tkzLabelPoints[below](X)
\tkzLabelPoints[left](A)
\tkzLabelPoints[right](B)
\tkzDrawCircle[R](O,4cm)
\node at (3,4) {$\mathscr{C}$};
\tkzDrawSegments(A,B X,Y)
\tkzDefPoint(20:4){M}
\tkzDrawPoints(M)
\tkzLabelPoints[above right](M)
\tkzDefLine[orthogonal=through M,K=-0.5](A,B) \tkzGetPoint{m}
\tkzDrawLine[color=gray!50](M,m)
\tkzInterLL(A,B)(M,m) \tkzGetPoint{D}
\tkzMarkRightAngle[color=gray!50](M,D,O)
\tkzLabelPoints[below right](O)
\tkzLabelPoints[below left](D)
\end{tikzpicture}
\end{center}

\item\par~
\begin{center}
\begin{tikzpicture}
\clip (-4.5,-4.5) rectangle (4.5,4.5);
\tkzDefPoint(0,0){O}
\tkzDefPoint(-4,0){A}
\tkzDefPoint(4,0){B}
\tkzDefPoint(0,-4){X}
\tkzDefPoint(0,4){Y}
\tkzDrawPoints(O)
\tkzLabelPoints[above](Y)
\tkzLabelPoints[below](X)
\tkzLabelPoints[left](A)
\tkzLabelPoints[right](B)
\tkzDrawCircle[R](O,4cm)
\node at (3,4) {$\mathscr{C}$};
\tkzDrawSegments(A,B X,Y)
\tkzDefPoint(20:4){M}
\tkzDrawPoints(M)
\tkzLabelPoints[above right](M)
\tkzDefLine[orthogonal=through M,K=-0.5](A,B) \tkzGetPoint{m}
\tkzDrawLine[color=gray!50](M,m)
\tkzInterLL(A,B)(M,m) \tkzGetPoint{D}
\tkzMarkRightAngle[color=gray!50](M,D,O)
\tkzLabelPoints[below right](O)
\tkzLabelPoints[below left](D)
%
\tkzDefLine[orthogonal=through M,K=0.5](X,Y) \tkzGetPoint{n}
\tkzDrawLine[color=gray!50](M,n)
\tkzInterLL(X,Y)(M,n) \tkzGetPoint{E}
\tkzMarkRightAngle[color=gray!50](M,E,O)
\tkzLabelPoints[above left](E)
\end{tikzpicture}
\end{center}
\end{enumerate}

\newpage

\item\par~
\begin{center}
\begin{tikzpicture}
\clip (-4.5,-4.5) rectangle (4.5,4.5);
\tkzDefPoint(0,0){O}
\tkzDefPoint(-4,0){A}
\tkzDefPoint(4,0){B}
\tkzDefPoint(0,-4){X}
\tkzDefPoint(0,4){Y}
\tkzDrawPoints(O)
\tkzLabelPoints[above](Y)
\tkzLabelPoints[below](X)
\tkzLabelPoints[left](A)
\tkzLabelPoints[right](B)
\tkzDrawCircle[R](O,4cm)
\node at (3,4) {$\mathscr{C}$};
\tkzDrawSegments(A,B X,Y)
\tkzDefPoint(20:4){M}
\tkzDrawPoints(M)
\tkzLabelPoints[above right](M)
\tkzDefLine[orthogonal=through M,K=-0.5](A,B) \tkzGetPoint{m}
\tkzDrawLine[color=gray!50](M,m)
\tkzInterLL(A,B)(M,m) \tkzGetPoint{D}
\tkzMarkRightAngle[color=gray!50](M,D,O)
\tkzLabelPoints[below right](O)
\tkzLabelPoints[below left](D)
%
\tkzDefLine[orthogonal=through M,K=0.5](X,Y) \tkzGetPoint{n}
\tkzDrawLine[color=gray!50](M,n)
\tkzInterLL(X,Y)(M,n) \tkzGetPoint{E}
\tkzMarkRightAngle[color=gray!50](M,E,O)
\tkzLabelPoints[above left](E)
\tkzDrawLine[color=gray!50](D,E)
\end{tikzpicture}
\end{center}

\item\par~
\begin{center}
\begin{tikzpicture}
\clip (-4.5,-4.5) rectangle (4.5,4.5);
\tkzDefPoint(0,0){O}
\tkzDefPoint(-4,0){A}
\tkzDefPoint(4,0){B}
\tkzDefPoint(0,-4){X}
\tkzDefPoint(0,4){Y}
\tkzDrawPoints(O)
\tkzLabelPoints[above](Y)
\tkzLabelPoints[below](X)
\tkzLabelPoints[left](A)
\tkzLabelPoints[right](B)
\tkzDrawCircle[R](O,4cm)
\node at (3,4) {$\mathscr{C}$};
\tkzDrawSegments(A,B X,Y)
\tkzDefPoint(20:4){M}
\tkzDrawPoints(M)
\tkzLabelPoints[above right](M)
\tkzDefLine[orthogonal=through M,K=-0.5](A,B) \tkzGetPoint{m}
\tkzDrawLine[color=gray!50](M,m)
\tkzInterLL(A,B)(M,m) \tkzGetPoint{D}
\tkzMarkRightAngle[color=gray!50](M,D,O)
\tkzLabelPoints[below right](O)
\tkzLabelPoints[below left](D)
%
\tkzDefLine[orthogonal=through M,K=0.5](X,Y) \tkzGetPoint{n}
\tkzDrawLine[color=gray!50](M,n)
\tkzInterLL(X,Y)(M,n) \tkzGetPoint{E}
\tkzMarkRightAngle[color=gray!50](M,E,O)
\tkzLabelPoints[above left](E)
\tkzDrawLine[color=gray!50](D,E)
\tkzDefLine[orthogonal=through M,K=0.5](D,E) \tkzGetPoint{p}
\tkzDrawLine[color=gray!50](M,p)
\tkzInterLL(D,E)(M,p) \tkzGetPoint{P}
\tkzMarkRightAngle[color=gray!50](M,P,E)
\tkzDrawPoints[color=red](P)
\end{tikzpicture}
\end{center}

\newpage

\item\par~
\begin{center}
\begin{tikzpicture}
\clip (-4.5,-4.5) rectangle (5,4.5);
\tkzDefPoint(0,0){O}
\tkzDefPoint(-4,0){A}
\tkzDefPoint(4,0){B}
\tkzDefPoint(0,-4){X}
\tkzDefPoint(0,4){Y}
\tkzDrawPoints(O)
\tkzLabelPoints[above](Y)
\tkzLabelPoints[below](X)
\tkzLabelPoints[left](A)
\tkzLabelPoints[right](B)
\tkzDrawCircle[R](O,4cm)
\tkzDrawSegments(A,B X,Y)
\tkzLabelPoints[below right](O)
\foreach\a in {20,30,40,50,60,70,80}
{
\tkzDefPoint(\a:4){M}
\tkzDrawPoints(M)
\tkzLabelPoint[above right](M){M$_{\a^\circ}$}
\tkzDefLine[orthogonal=through M,K=-0.5](A,B) \tkzGetPoint{m}
\tkzDrawLine[color=gray!20](M,m)
\tkzInterLL(A,B)(M,m) \tkzGetPoint{D}
\tkzMarkRightAngle[color=gray!50](M,D,O)
\tkzDefLine[orthogonal=through M,K=0.5](X,Y) \tkzGetPoint{n}
\tkzDrawLine[color=gray!20](M,n)
\tkzInterLL(X,Y)(M,n) \tkzGetPoint{E}
\tkzMarkRightAngle[color=gray!50](M,E,O)
\tkzDrawLine[color=gray!20](D,E)
\tkzDefLine[orthogonal=through M,K=0.5](D,E) \tkzGetPoint{p}
\tkzDrawLine[color=gray!20](M,p)
\tkzInterLL(D,E)(M,p) \tkzGetPoint{P}
\tkzMarkRightAngle[color=gray!20](M,P,E)
\tkzDrawPoints[color=red](P)
}
\foreach\a in {100,110,120,130,140,150,160,170,190,200,210,220,230, 240,250,260,280,290,300,310,320,330,340,350}
{
\tkzDefPoint(\a:4){M}
\tkzDefLine[orthogonal=through M,K=-0.5](A,B) \tkzGetPoint{m}
\tkzInterLL(A,B)(M,m) \tkzGetPoint{D}
\tkzDefLine[orthogonal=through M,K=0.5](X,Y) \tkzGetPoint{n}
\tkzInterLL(X,Y)(M,n) \tkzGetPoint{E}
\tkzDefLine[orthogonal=through M,K=0.5](D,E) \tkzGetPoint{p}
\tkzInterLL(D,E)(M,p) \tkzGetPoint{P}
\tkzDrawPoints[color=orange](P)
}
\end{tikzpicture}
\end{center}
\end{enumerate}
\end{document}