ABCDEFGH designates a cube with side length 1.\\
Point I is the midpoint of segment [BF]. Point J is the midpoint of segment [BC]. Point K is the midpoint of segment [CD].

\section*{Part A}
\textit{In this part, no justification is required.}\\
We admit that the lines (IJ) and (CG) intersect at a point L.\\
Construct, on the figure provided in the appendix and leaving the construction lines visible:
\begin{itemize}
  \item the point L;
  \item the intersection $\mathscr { D }$ of the planes (IJK) and (CDH);
  \item the cross-section of the cube by the plane (IJK).
\end{itemize}

\section*{Part B}
Space is referred to the coordinate system ( $\mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } }$ ).

\begin{enumerate}
  \item Give the coordinates of $\mathrm { A } , \mathrm { G } , \mathrm { I } , \mathrm { J }$ and K in this coordinate system.
  \item a. Show that the vector $\overrightarrow { \mathrm { AG } }$ is normal to the plane (IJK).\\
b. Deduce a Cartesian equation of the plane (IJK).
  \item We denote by $M$ a point of the segment [AG] and $t$ the real number in the interval $[ 0 ; 1 ]$ such that $\overrightarrow { \mathrm { AM } } = t \overrightarrow { \mathrm { AG } }$.\\
a. Prove that $M \mathrm { I } ^ { 2 } = 3 t ^ { 2 } - 3 t + \frac { 5 } { 4 }$.\\
b. Prove that the distance $M I$ is minimal for the point $\mathrm { N } \left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$.
  \item Prove that for this point $\mathrm { N } \left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$:\\
a. N belongs to the plane (IJK).\\
b. The line (IN) is perpendicular to the lines (AG) and (BF).
\end{enumerate}