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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 1316512775804--
%%--Calculs vectoriels--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
On consid\`{e}re un triangle OAB.

\begin{enumerate}
\item Montrer que le point $I$ d\'{e}fini par $\overrightarrow{OI}=\frac{1}%
{2}(\overrightarrow{OA}+\overrightarrow{OB})$ est le milieu de $\left[
AB\right]  ;$

\item Construire les points $M$ et $N$ tels que:%
\[
\overrightarrow{OM}+\overrightarrow{ON}=\overrightarrow{OA}\quad\text{et}%
\quad\overrightarrow{OM}-\overrightarrow{ON}=\overrightarrow{OB};
\]


\item Soit $J$ le milieu de $\left[  OA\right]  ;$ d\'{e}montrer que $J$ est
aussi le milieu de $\left[  MN\right]  .$

\item Construire un parall\'{e}logramme $ABCD$ et les points $I$ et $J$
d\'{e}finis par:%
\[
\overrightarrow{BI}=\frac{1}{2}\overrightarrow{AB}\quad\text{et}%
\quad\overrightarrow{AJ}=3\overrightarrow{AD};
\]


\item Montrer que:%
\[
\overrightarrow{IJ}=-\dfrac{3}{2}\overrightarrow{AB}+3\overrightarrow{AD}%
\quad\text{et}\quad\overrightarrow{IC}=-\dfrac{1}{2}\overrightarrow
{AB}+\overrightarrow{AD};
\]


\item Prouver que $\overrightarrow{IJ}$ et $\overrightarrow{IC}$ sont colin\'{e}aires.

Que peut-on en conclure pour les points $I,$ $J$ et $C.$
\end{enumerate}
\end{exercise}


\end{document}