Let ABCDEFGH be a cube and I the center of the square ADHE, that is, the midpoint of segment [AH] and segment [ED]. Let J be a point on segment [CG].\\
The cross-section of the cube ABCDEFGH by the plane (FIJ) is the quadrilateral FKLJ.

We place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.\\
We have therefore $\mathrm{A}(0;0;0)$, $\mathrm{B}(1;0;0)$, $\mathrm{D}(0;1;0)$ and $\mathrm{E}(0;0;1)$.\\
Parts A and B can be treated independently.

\section*{Part A}
In this part, the point J has coordinates $\left(1; 1; \frac{2}{5}\right)$.

\begin{enumerate}
  \item Prove that the coordinates of point I are $\left(0; \frac{1}{2}; \frac{1}{2}\right)$.
  \item a. Prove that the vector $\vec{n}\begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix}$ is a normal vector to the plane (FIJ).\\
b. Prove that a Cartesian equation of the plane (FIJ) is
$$-x + 3y + 5z - 4 = 0.$$
  \item Let $d$ be the line perpendicular to the plane (FIJ) and passing through B.\\
a. Determine a parametric representation of the line $d$.\\
b. We denote by M the point of intersection of the line $d$ and the plane (FIJ). Prove that $\mathrm{M}\left(\frac{6}{7}; \frac{3}{7}; \frac{5}{7}\right)$.
  \item a. Calculate $\overrightarrow{\mathrm{BM}} \cdot \overrightarrow{\mathrm{BF}}$.\\
b. Deduce an approximate value to the nearest degree of the angle $\widehat{\mathrm{MBF}}$.
\end{enumerate}

\section*{Part B}
In this part, J is an arbitrary point on segment [CG].\\
Its coordinates are therefore $(1; 1; a)$, where $a$ is a real number in the interval $[0; 1]$.

\begin{enumerate}
  \item Show that the cross-section of the cube by the plane (FIJ) is a parallelogram.
  \item We admit that L has coordinates $\left(0; 1; \frac{a}{2}\right)$.\\
For which value(s) of $a$ is the quadrilateral FKLJ a rhombus?
\end{enumerate}