\section*{Exercise 4 (For candidates who have not followed the specialty course)}
We consider a cube $ABCDEFGH$ with edge length 1. We denote $I$ the midpoint of segment $[EF]$, $J$ the midpoint of segment $[EH]$ and $K$ the point of segment $[AD]$ such that $\overrightarrow{AK} = \frac{1}{4}\overrightarrow{AD}$.\\
We denote $\mathscr{P}$ the plane passing through $I$ and parallel to the plane $(FHK)$.

\section*{Part A}
In this part, the constructions requested will be performed without justification on the figure given in the appendix.
\begin{enumerate}
  \item The plane $(FHK)$ intersects the line $(AE)$ at a point which we denote $M$. Construct the point $M$.
  \item Construct the cross-section of the cube by the plane $\mathscr{P}$.
\end{enumerate}

\section*{Part B}
In this part, we equip the space with the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE})$.\\
We recall that $\mathscr{P}$ is the plane passing through $I$ and parallel to the plane $(FHK)$.

\begin{enumerate}
  \item a. Show that the vector $\vec{n}\left(\begin{array}{c} 4 \\ 4 \\ -3 \end{array}\right)$ is a normal vector to the plane $(FHK)$.\\
  b. Deduce that a Cartesian equation of the plane $(FHK)$ is: $4x + 4y - 3z - 1 = 0$.\\
  c. Determine an equation of the plane $\mathscr{P}$.\\
  d. Calculate the coordinates of the point $M'$, the point of intersection of the plane $\mathscr{P}$ and the line $(AE)$.
  \item We denote $\Delta$ the line passing through point $E$ and perpendicular to the plane $\mathscr{P}$.\\
  a. Determine a parametric representation of the line $\Delta$.\\
  b. Calculate the coordinates of point $L$, the intersection of line $\Delta$ and plane $(ABC)$.\\
  c. Draw the line $\Delta$ on the figure provided in the appendix.\\
  d. Are the lines $\Delta$ and $(BF)$ intersecting? What about the lines $\Delta$ and $(CG)$? Justify.
\end{enumerate}