\documentclass{beamer}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Fichier de style de T. Duvoid -- version du 14/11/2007%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usetheme{progressbar}
%Les extensions utilis\'ees

\usepackage{pstricks}
\usepackage{pst-plot,pstricks-add,pst-math,pst-xkey,pst-eucl,pst-tree,pst-node,} %manque picins
\usepackage[frenchb]{babel}
\usepackage[utf8]{inputenc}



%Cr\'eation des ensembles math\'ematiques
\newcommand{\N}{\ensuremath{\mbox{I\hspace{-.15em}N}}\xspace}
\newcommand{\R}{\ensuremath{\mbox{I\hspace{-.15em}R}}\xspace}
\newcommand{\C}{\ensuremath{\mbox{l\hspace{-.47em}C}}\xspace}
\newcommand{\Z}{\ensuremath{\mbox{Z\hspace{-.3em}Z}}\xspace}
\newcommand{\Q}{\ensuremath{\mbox{l\hspace{-.47em}Q}}\xspace}
\newcommand{\D}{\ensuremath{\mbox{I\hspace{-.15em}D}}\xspace}

%Arc de cercle 
 \newlength{\longarc} 
 \newcommand{\arc}[1]{\settowidth{\longarc}{$#1$}% 
         \addtolength{\longarc}{-0.5em}% 
 \unitlength \longarc% 
 \ensuremath{% 
         \stackrel{\begin{picture}(1,0.2) 
                 \qbezier(0,0)(0.5,0.2)(1,0) 
         \end{picture}} 
         {#1} 
 }}
%flèches d'\'equivalence et de vecteurs
\newcommand{\ssi}{\ensuremath{\Longleftrightarrow}}
\newcommand{\vecteur}[1]{\overrightarrow{#1\;}}


%
%%%%%%%%%%%
%Symbole Attention
\font\manual=manfnt
\newcommand{\danger}{{\manual\symbol{127}}}
%%%%%%%%Pour des flËches dans les tableaux de variation
\newcommand{\cro}{% croissante
\unitlength=1cm
\begin{minipage} {1cm}%pour centrer verticalement f(x)
  \begin{picture}(1,1)    % dessin de 2 X 2
     \put(0,0){\vector(1,1){1}} % 1 ‡ partir (0,0) direction (1,1)  
 \end{picture} 
\end{minipage}
}
\newcommand{\dec}{%dÈcroissante
\unitlength=1cm
\begin{minipage} {1cm}%pour centrer verticalement f(x)
  \begin{picture}(1,1)    % dessin de 2 X 2
     \put(0,1){\vector(1,-1){1}} 
 \end{picture} 
\end{minipage}
}
%%toutes les maths en style math
\everymath{\displaystyle}
%%
\newcommand{\repere}{$(O;\vec{\imath},\vec{\jmath})$}
%%
\newcounter{numex} 
\def\Ex#1{\addtocounter{numex}{1} \mbox{\large{\textsc{Exercice}} 
\thenumex { \scriptsize #1}}}

%\def \points#1%
%{%
%\if -#1
%\xspace%
% \else
%{\small ( #1 points )}\xspace%
%\fi
%}
%pour compatibilitÈ ancienne version
\def\exop#1{\exo[#1]}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareMathOperator{\e}{e}




\title{Réciproque du théorème de Thalès}
\author{~}
\begin{document}

\begin{frame}{Parallèles ?}
	Les droites $(LA)$ et $(HT)$ sont-elles parallèles ?
	\begin{center}
		\psset{xunit=1.0cm,yunit=1.0cm,algebraic=true,dotstyle=o,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
		\begin{pspicture*}(-2,-0.3)(3,3.5)
		\psplot{-2}{3}{(--7.52-2.5*x)/3.44}
		\psplot{-2}{3}{(--2.55--1.63*x)/1.83}
		\psplot{-2}{3}{(--0.96--0.14*x)/3.88}
		\psplot{-2}{3}{(--11.15--0.14*x)/3.88}
		\rput[bl](-0.82,2.96){$H$}
		\rput[bl](2.62,0.46){$A$}
		\rput[bl](-1.54,0.5){$L$}
		\rput[bl](0.38,2.06){$M$}
		\rput[bl](1.62,3.08){$T$}
		\rput[bl](-0.5,2.06){$4$}
		\rput[bl](-0.7,1.06){$6$}
		\rput[bl](1.2,2.06){$8$}
		\rput[bl](1.6,1.1){$3$}
		\end{pspicture*}
	\end{center}
\end{frame}
	

\begin{frame}{Rapports}
	\begin{center}
		\begin{tikzpicture}[scale=0.7][line cap=round,line join=round,x=1.0cm,y=1.0cm]
		\clip(-2.3,-2) rectangle (2.2,4);
		\draw [domain=-2.3:2.2] plot(\x,{(--2.21--2.42*\x)/1.82});
		\draw [domain=-2.3:2.2] plot(\x,{(-2.88--2.12*\x)/-1.34});
		\draw [domain=-2.3:2.2] plot(\x,{(-2.01--0.3*\x)/3.16});
		\draw [domain=-2.3:2.2] plot(\x,{(--0.35--0.3*\x)/3.16});
		\draw [domain=-2.3:2.2] plot(\x,{(--9.82--0.3*\x)/3.16});
		\draw (-1.5,-0.78)-- (-0.89,0.03);
		\draw (-1.29,-0.35) -- (-1.15,-0.46);
		\draw (-1.25,-0.3) -- (-1.1,-0.4);
		\draw (0.32,1.64)-- (-0.29,0.83);
		\draw (0.11,1.21) -- (-0.03,1.32);
		\draw (0.07,1.16) -- (-0.08,1.26);
		\draw (0.32,1.64)-- (0.77,0.94);
		\draw (0.6,1.2) -- (0.43,1.22);
		\draw (0.6,1.2) -- (0.66,1.36);
		\draw (0.77,0.94)-- (1.21,0.23);
		\draw (1.05,0.49) -- (0.88,0.51);
		\draw (1.05,0.49) -- (1.1,0.65);
		\draw (1.21,0.23)-- (1.66,-0.48);
		\draw (1.49,-0.22) -- (1.32,-0.2);
		\draw (1.49,-0.22) -- (1.55,-0.06);
		\draw (0.32,1.64)-- (0.93,2.45);
		\draw (0.53,2.07) -- (0.68,1.96);
		\draw (0.57,2.13) -- (0.72,2.02);
		\draw (0.93,2.45)-- (1.53,3.25);
		\draw (1.14,2.88) -- (1.28,2.77);
		\draw (1.18,2.93) -- (1.32,2.83);
		\draw (0.32,1.64)-- (-0.13,2.35);
		\draw (0.04,2.08) -- (0.21,2.07);
		\draw (0.04,2.08) -- (-0.02,1.92);
		\draw (-0.13,2.35)-- (-0.57,3.05);
		\draw (-0.41,2.79) -- (-0.24,2.77);
		\draw (-0.41,2.79) -- (-0.47,2.63);
		\draw (-0.89,0.03)-- (-0.29,0.83);
		\draw (-0.68,0.46) -- (-0.54,0.35);
		\draw (-0.64,0.51) -- (-0.5,0.4);
		\draw  (-1.5,-0.78)-- ++(-0.5pt,-0.5pt) -- ++(1.0pt,1.0pt) ++(-1.0pt,0) -- ++(1.0pt,-1.0pt);
		\draw (-1.42,-1.02) node {$A$};
		\draw  (1.66,-0.48)-- ++(-0.5pt,-0.5pt) -- ++(1.0pt,1.0pt) ++(-1.0pt,0) -- ++(1.0pt,-1.0pt);
		\draw (1.52,-0.8) node {$B$};
		\draw  (0.32,1.64)-- ++(-0.5pt,-0.5pt) -- ++(1.0pt,1.0pt) ++(-1.0pt,0) -- ++(1.0pt,-1.0pt);
		\draw (0.56,1.62) node {$O$};
		\draw  (1.53,3.25)-- ++(-0.5pt,-0.5pt) -- ++(1.0pt,1.0pt) ++(-1.0pt,0) -- ++(1.0pt,-1.0pt);
		\draw (1.86,3.06) node {$N_2$};
		\draw  (-0.89,0.03)-- ++(-0.5pt,-0.5pt) -- ++(1.0pt,1.0pt) ++(-1.0pt,0) -- ++(1.0pt,-1.0pt);
		\draw (-0.72,-0.24) node {$N_1$};
		\draw (-0.72,2.84) node {$M_2$};
		\draw  (1.21,0.23)-- ++(-0.5pt,-0.5pt) -- ++(1.0pt,1.0pt) ++(-1.0pt,0) -- ++(1.0pt,-1.0pt);
		\draw (1.1,-0.04) node {$M_1$};
		\draw  (-0.29,0.83)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
		\draw  (0.93,2.45)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
		\draw  (-0.13,2.35)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
		\draw  (0.77,0.94)-- ++(-1.5pt,-1.5pt) -- ++(3.0pt,3.0pt) ++(-3.0pt,0) -- ++(3.0pt,-3.0pt);
		\end{tikzpicture}
	\end{center}
	Déterminer la valeur des rapports $\frac{OM_1}{OB}$, $\frac{ON_1}{OA}$ et $\frac{ON_2}{OA}$.\\
	Que pouvez vous en conclure pour les droites $(M_1N_1)$ et $(AB)$ ?\\
	Que pouvez vous en conclure pour les droites $(M_1N_2)$ et $(AB)$ ?\\
\end{frame}
	



\begin{frame}{Triangle DFG}
	$DFG$ est un triangle tel que $DF=11$ cm, $FG=6$ cm,  $DG=7,7$ cm, $a \in [DF]$, $DA=9$ cm, $E \in [DG]$ et $DE=6,3$ cm.\\
	\vspace{1cm}
	Démontrer que les droites $(AE)$ et $(FG)$ sont parallèles.
\end{frame}

\begin{frame}{Triangle DCV}
	$DCV$ est un triangle tel que $DV=6,4$ cm, $DC=3,6$ cm,  $CV=4$ cm, $D \in [VO]$, $DO=5,5$ cm, $D \in [CA]$ et $DA=3,1$ cm.\\
	\vspace{1cm}
	 Déterminer la position relative des droites $(CV)$ et $(AO)$.
\end{frame}



\begin{frame}{Quadrilatère}
	$ABCD$ est un quadrilatère. $O$ est le point d'intersection de ses diagonales.\\
	$OA=2,4$ cm, $OB=3,6$cm, $OC=2,8$ cm, $OD=4,2$ cm, $AB=4,2$ cm.\\
	\vspace{1cm}
	
	Calculer la longueur $DC$.
\end{frame}


\begin{frame}{Incontournable ?}
	$ABC$ est un triangle rectangle en $A$ tel que $AB=12$ cm et $AC=8$ cm. $F$ est le point du segment $[AC]$ tel que $AF=4$ cm et $E$ le point du segment $[AB]$ tel que $AE=6$ cm.\\
	\vspace{1cm}
 Démontrer que les droites $(EF)$ et $(BC)$ sont parallèles.
\end{frame}



\begin{frame}{Contrefort}
	\begin{minipage}{6cm}
	Pour consolider un bâtiment, des charpentiers ont construit un contrefort en bois schématisé ci-dessous. Les mesures sont exprimées en mètres.
	\begin{enumerate}
		\item En considérant que le montant $[BS]$ est perpendiculaire au sol, calculer la longueur $AS$.
		\item Calculer les longueurs $SM$ et $SN$.
		\item Démontrer que la traverse $[MN]$ est bien parallèle au sol.
	\end{enumerate}
\end{minipage}
\begin{minipage}{4cm}
	\begin{center}
	\begin{tikzpicture}
	\coordinate(A) at (0,0);
	\coordinate(B) at (2,0);
	\coordinate(C) at (4,0);
	\coordinate(M) at (0.7,1.8);
	\coordinate(N) at (2,1.8);
	\coordinate(S) at (2,5);
	\coordinate(T) at (2,6);
	\coordinate(U) at (4,6);
	\draw (B) -- (C) -- (U) -- (T) -- cycle;
	\draw (S) -- (A);
	\draw (A)-- (B);
	\draw (M)-- (N);
	\draw (B) -- (C) node[midway, below]{$sol$};
	\draw (A) node[below left] {$A$};
	\draw (B) node[below] {$B$};
	\draw (M) node[above left] {$M$};
	\draw (N) node[above right] {$N$};
	\draw (S) node[left] {$S$};
	\draw[<->] (-.1,0) -- (0.6,1.8) node[midway, left] {$1.95$};
	\draw[<->] (0,-0.1) -- (2,-0.1) node[midway, below] {$2.5$};
	\draw[<->] (2.1,0) -- (2.1,1.8) node[midway, right] {$1.8$};
	\draw[<->] (3,0) -- (3,5) node[midway, right] {$6$};
	\end{tikzpicture}
	\end{center}
\end{minipage}
\end{frame}
\end{document}