\section*{Exercise 3 (4 points)}

We consider a cube ABCDEFCH given in Appendix 2 (to be returned with your work).\\
We denote M the midpoint of segment $[\mathrm{EH}]$, N that of $[\mathrm{FC}]$ and P the point such that
$\overrightarrow{\mathrm{HP}} = \frac{1}{4} \overrightarrow{\mathrm{HG}}$.

\section*{Part A: Section of the cube by the plane (MNP)}
\begin{enumerate}
  \item Justify that the lines (MP) and (FG) are secant at a point L.
\end{enumerate}

Construct the point L.\\
2. We admit that the lines (LN) and (CG) are secant and we denote T their point of intersection.

We admit that the lines (LN) and (BF) are secant and we denote Q their point of intersection.\\
a. Construct the points T and Q leaving the construction lines visible.\\
b. Construct the intersection of the planes (MNP) and (ABF).\\
3. Deduce a construction of the section of the cube by the plane (MNP).

\section*{Part B}
The 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 points $\mathrm{M}, \mathrm{N}$ and P in this coordinate system.
  \item Determine the coordinates of point L.
  \item We admit that point T has coordinates $\left(1 ; 1 ; \frac{5}{8}\right)$. Is the triangle TPN right-angled at T?
\end{enumerate}