\documentclass[a4paper,9pt,french]{article}


\input macro_dwicky_final.tex
\geometry{verbose,letterpaper,tmargin=0.5cm,bmargin=0.5cm,lmargin=2cm,rmargin=2cm}

\begin{document}
\pagestyle{empty}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	Cadre jaune
\textbf{Nom : }$\dots\dots\dots\dots\dots$ \hfill \textbf{Pr\'enom : }$\dots\dots\dots\dots\dots$  \hfill Classe :$\dots\dots$\\
\psframebox[linestyle=solid,shadow=true,framearc=0.5,fillcolor=vert!30]{
\begin{minipage}{17cm}
\begin{center}
\vspace{1mm}
\textsc{\textbf{ Interrogation n$^{\circ}9$ }}
\end{center}
\vspace{-3mm}
\end{minipage}
}
\setcounter{section}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	 Interro
\begin{center}
\textit{\textbf{On prendra soin de coller le sujet sur la copie. La note tiendra compte de la qualité de la rédaction et de l'application.}}
\end{center}

\Exoi{\textit{Question de cours}\hfill (2 points)\\
Compléter :\\
\parbox[]{10cm}{
\Thc{
	Si $P$ et $Q$ sont deux plans $\dots\dots\dots\dots\dots$, alors tout plan  qui 
	coupe $P$ $ \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots$ et les droites d'intersection $ \dots\dots\dots\dots\dots$
}
                 }
\parbox[]{6cm}{\begin{center}\includegraphics[scale=1]{fig2c_espace.8}\end{center}}}

\Exoi{\textit{Section d'un solide par un plan}\hfill (4 points)\\
Dans cet exercice on ne demande aucune justification.
\begin{figure}[h!]
   \begin{minipage}[b]{0.40\linewidth}
   On a $I\in(SAC) $,  $J\in(SAC)$ et $K\in[BS]$  
      \centering \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-1.72,-1.02) rectangle (5.06,4.98);
\draw (1.34,4.36)-- (-0.94,0.16);
\draw (1.34,4.36)-- (3.96,0.28);
\draw [dotted] (-0.94,0.16)-- (3.96,0.28);
\draw (3.96,0.28)-- (2.04,-0.4);
\draw (-0.94,0.16)-- (2.04,-0.4);
\draw (2.04,-0.4)-- (1.34,4.36);
\fill [color=black] (-0.94,0.16) circle (0.5pt);
\draw[color=black] (-1.3,0.26) node {$A$};
\fill [color=black] (3.96,0.28) circle (0.5pt);
\draw[color=black] (4.1,0.46) node {$B$};
\fill [color=black] (2.04,-0.4) circle (0.5pt);
\draw[color=black] (1.94,-0.64) node {$C$};
\fill [color=black] (1.34,4.36) circle (0.5pt);
\draw[color=black] (1.32,4.7) node {$S$};
\draw [color=black] (2.63,2.36)-- ++(-1.0pt,-1.0pt) -- ++(2.0pt,2.0pt) ++(-2.0pt,0) -- ++(2.0pt,-2.0pt);
\draw[color=black] (2.78,2.58) node {$K$};
\draw [color=black] (0.26,1.74)-- ++(-1.0pt,-1.0pt) -- ++(2.0pt,2.0pt) ++(-2.0pt,0) -- ++(2.0pt,-2.0pt);
\draw[color=black] (0.36,1.96) node {$I$};
\draw [color=black] (1.18,0.92)-- ++(-1.0pt,-1.0pt) -- ++(2.0pt,2.0pt) ++(-2.0pt,0) -- ++(2.0pt,-2.0pt);
\draw[color=black] (1.3,1.14) node {$J$};
\end{tikzpicture}
      \caption{\it Dessiner la section du tétraèdre $SABC$ et du plan $(IJK)$ }
   \end{minipage}\hfill
   \begin{minipage}[b]{0.48\linewidth} 
   On a $I\in(ABC) $,  $J\in(SAB)$ et $K\in[BS]$  
      \centering\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(0.7,-1.4) rectangle (7.64,4.9);
\draw (1.48,-0.1)-- (5.02,-0.92);
\draw (5.02,-0.92)-- (7.02,-0.14);
\draw [dotted] (7.02,-0.14)-- (1.48,-0.1);
\draw (1.48,-0.1)-- (4.34,4.3);
\draw (4.34,4.3)-- (7.02,-0.14);
\draw (5.02,-0.92)-- (4.34,4.3);
\fill [color=black] (1.48,-0.1) circle (0.5pt);
\draw[color=black] (1.14,-0.02) node {$A$};
\fill [color=black] (7.02,-0.14) circle (0.5pt);
\draw[color=black] (7.18,0.06) node {$B$};
\fill [color=black] (5.02,-0.92) circle (0.5pt);
\draw[color=black] (5.2,-1.12) node {$C$};
\fill [color=black] (4.34,4.3) circle (0.5pt);
\draw[color=black] (4.38,4.64) node {$S$};
\fill [color=black] (4.32,-0.52) circle (0.5pt);
\draw[color=black] (4.46,-0.26) node {$I$};
\fill [color=black] (4.14,0.84) circle (0.5pt);
\draw[color=black] (4.06,1.14) node {$J$};
\fill [color=black] (5.74,1.98) circle (0.5pt);
\draw[color=black] (5.9,2.18) node {$K$};
\end{tikzpicture}
      \caption{\it Dessiner la section du tétraèdre $SABC$ et du plan $(IJK)$ }
   \end{minipage}
   \end{figure}
   \begin{figure}[h!]
   \begin{minipage}[b]{0.40\linewidth}
  On a $I\in(EFG)$, $J\in[HG]$ et $K\in[DC]$
      \centering  \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(1.88,-1.86) rectangle (8.36,3.98);
\draw (2.86,-1.36)-- (6.94,-1.42);
\draw (6.94,-1.42)-- (7,2.66);
\draw (7,2.66)-- (2.92,2.72);
\draw (2.92,2.72)-- (2.86,-1.36);
\draw [dotted] (2.86,-1.36)-- (3.72,-0.52);
\draw (2.92,2.72)-- (3.78,3.56);
\draw (3.78,3.56)-- (7.86,3.5);
\draw (7.86,3.5)-- (7,2.66);
\draw (7.86,3.5)-- (7.8,-0.58);
\draw [dotted] (7.8,-0.58)-- (3.72,-0.52);
\draw (6.94,-1.42)-- (7.8,-0.58);
\draw [dotted] (3.78,3.56)-- (3.72,-0.52);
\fill [color=black] (2.86,-1.36) circle (0.5pt);
\draw[color=black] (2.62,-1.3) node {$A$};
\fill [color=black] (6.94,-1.42) circle (0.5pt);
\draw[color=black] (7.24,-1.46) node {$B$};
\fill [color=black] (2.92,2.72) circle (0.5pt);
\draw[color=black] (2.7,2.9) node {$D$};
\fill [color=black] (7,2.66) circle (0.5pt);
\draw[color=black] (6.92,2.98) node {$C$};
\fill [color=black] (3.72,-0.52) circle (0.5pt);
\draw[color=black] (3.88,-0.32) node {$E$};
\fill [color=black] (7.8,-0.58) circle (0.5pt);
\draw[color=black] (7.94,-0.38) node {$F$};
\fill [color=black] (7.86,3.5) circle (0.5pt);
\draw[color=black] (8.02,3.7) node {$G$};
\fill [color=black] (3.78,3.56) circle (0.5pt);
\draw[color=black] (3.94,3.76) node {$H$};
\fill [color=black] (5.26,0.4) circle (0.5pt);
\draw[color=black] (5.46,0.42) node {$I$};
\fill [color=black] (5.32,3.54) circle (0.5pt);
\draw[color=black] (5.46,3.74) node {$J$};
\fill [color=black] (6.32,2.67) circle (0.5pt);
\draw[color=black] (6.48,2.86) node {$K$};
\end{tikzpicture}
      \caption{\it Dessiner la section du cube $ABCDEFGH$ et du plan $(IJK)$ }
   \end{minipage}\hfill
   \begin{minipage}[b]{0.48\linewidth}   
      \centering $I\in(ABC)$, $ J\in (DCG)$ et $K\in[DC]$ 
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(2.3,-1.8) rectangle (8.5,3.88);
\draw (2.86,-1.36)-- (6.94,-1.42);
\draw (6.94,-1.42)-- (7,2.66);
\draw (7,2.66)-- (2.92,2.72);
\draw (2.92,2.72)-- (2.86,-1.36);
\draw [dotted] (2.86,-1.36)-- (3.72,-0.52);
\draw (2.92,2.72)-- (3.78,3.56);
\draw (3.78,3.56)-- (7.86,3.5);
\draw (7.86,3.5)-- (7,2.66);
\draw (7.86,3.5)-- (7.8,-0.58);
\draw [dotted] (7.8,-0.58)-- (3.72,-0.52);
\draw (6.94,-1.42)-- (7.8,-0.58);
\draw [dotted] (3.78,3.56)-- (3.72,-0.52);
\fill [color=black] (2.86,-1.36) circle (0.5pt);
\draw[color=black] (2.62,-1.3) node {$A$};
\fill [color=black] (6.94,-1.42) circle (0.5pt);
\draw[color=black] (7.24,-1.46) node {$B$};
\fill [color=black] (2.92,2.72) circle (0.5pt);
\draw[color=black] (2.7,2.9) node {$D$};
\fill [color=black] (7,2.66) circle (0.5pt);
\draw[color=black] (6.92,2.98) node {$C$};
\fill [color=black] (3.72,-0.52) circle (0.5pt);
\draw[color=black] (3.88,-0.32) node {$E$};
\fill [color=black] (7.8,-0.58) circle (0.5pt);
\draw[color=black] (7.94,-0.38) node {$F$};
\fill [color=black] (7.86,3.5) circle (0.5pt);
\draw[color=black] (8.02,3.7) node {$G$};
\fill [color=black] (3.78,3.56) circle (0.5pt);
\draw[color=black] (3.94,3.76) node {$H$};
\fill [color=black] (5.26,0.4) circle (0.5pt);
\draw[color=black] (5.46,0.42) node {$I$};
\fill [color=black] (5.48,3.04) circle (0.5pt);
\draw[color=black] (5.62,3.24) node {$J$};
\fill [color=black] (6.32,2.67) circle (0.5pt);
\draw[color=black] (6.48,2.86) node {$K$};
\end{tikzpicture}
      \caption{\it Dessiner la section du cube $ABCDEFGH$ avec le plan $(IJK)$ }
   \end{minipage}
\end{figure}
}
\pagebreak

\Exoi{\textit{Raisonner dans l'espace}\hfill(4 points)\\
$ABCDEFGH$ est un cube.\\
$I$, $J$, $K$ et $L$ sont les milieux respectifs des segments $[AD]$, $[AB]$, $[BF]$ et $[FG]$. 
\begin{enumerate}
\item Déterminer l'intersection des plans $(ABC)$ et $(BFG)$.
\item Quelle est la nature du quadrilatère $AILF$ ? En déduire que $(IL)$ et $(AF)$ sont parallèles.
\item Démontrer que les droites $(AF)$ et $(JK)$ sont parallèles (on utilisera le théorème des milieux), puis en déduire que $(JK)$ et $(AF)$ sont parallèles.
\item En déduire que les droites $(IJ)$ et $(KL)$ sont coplanaires, et donc sécantes en un point $S$.
\end{enumerate}
}


\begin{center}

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\begin{pspicture*}(2.16,-2.18)(8.52,4)
\psline(2.86,-1.36)(6.94,-1.42)
\psline(6.94,-1.42)(7,2.66)
\psline(7,2.66)(2.92,2.72)
\psline(2.92,2.72)(2.86,-1.36)
\psline[linestyle=dotted](2.86,-1.36)(3.72,-0.52)
\psline(2.92,2.72)(3.78,3.56)
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\psplot{2.89}{8.52}{(--7.35-2.07*x)/2.01}
\psplot{2.16}{7.83}{(--18.59-2.46*x)/-0.46}
\psdots[dotsize=1pt 0,dotstyle=*](2.86,-1.36)
\rput[bl](2.54,-1.46){$A$}
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\psdots[dotsize=1pt 0,dotstyle=*](2.92,2.72)
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\psdots[dotsize=1pt 0,dotstyle=*](7.83,1.46)
\rput[bl](7.92,1.5){$L$}
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\end{center}



\end{document}
