We consider the cube ABCDEFGH.\\
We are given three points I, J and K satisfying:
$$\overrightarrow { \mathrm { EI } } = \frac { 1 } { 4 } \overrightarrow { \mathrm { EH } } , \quad \overrightarrow { \mathrm { EJ } } = \frac { 1 } { 4 } \overrightarrow { \mathrm { EF } } , \quad \overrightarrow { \mathrm { BK } } = \frac { 1 } { 4 } \overrightarrow { \mathrm { BF } }$$
We use the orthonormal coordinate system $(A ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } })$.

\begin{enumerate}
  \item Give without justification the coordinates of points I, J and K.
  \item Prove that the vector $\overrightarrow { \mathrm { AG } }$ is normal to the plane (IJK).
  \item Show that a Cartesian equation of the plane (IJK) is $4 x + 4 y + 4 z - 5 = 0$.
  \item Determine a parametric representation of the line (BC).
  \item Deduce the coordinates of point L, the point of intersection of the line (BC) with the plane (IJK).
  \item On the figure in the appendix, place point L and construct the intersection of the plane (IJK) with the face (BCGF).
  \item Let $\mathrm { M } \left( \frac { 1 } { 4 } ; 1 ; 0 \right)$. Show that the points I, J, L and M are coplanar.
\end{enumerate}