\textbf{Exercise 4 — Candidates who have not followed the specialization course}

In space, we consider a tetrahedron ABCD whose faces ABC, ACD and ABD are right-angled and isosceles triangles at A. We denote by E, F and G the midpoints of sides $[\mathrm{AB}]$, $[\mathrm{BC}]$ and $[\mathrm{CA}]$ respectively.\\
We choose AB as the unit of length and we place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AC}}, \overrightarrow{\mathrm{AD}})$ of space.

\begin{enumerate}
  \item We denote by $\mathscr { P }$ the plane that passes through A and is perpendicular to the line $(\mathrm{DF})$.\\
We denote by H the point of intersection of plane $\mathscr { P }$ and line (DF).\\
a. Give the coordinates of points D and F.\\
b. Give a parametric representation of line (DF).\\
c. Determine a Cartesian equation of plane $\mathscr { P }$.\\
d. Calculate the coordinates of point H.\\
e. Prove that the angle $\widehat{\mathrm{EHG}}$ is a right angle.\\
  \item We denote by $M$ a point on line (DF) and by $t$ the real number such that $\overrightarrow{\mathrm{DM}} = t \overrightarrow{\mathrm{DF}}$. We denote by $\alpha$ the measure in radians of the geometric angle $\widehat{\mathrm{EMG}}$.\\
The purpose of this question is to determine the position of point $M$ so that $\alpha$ is maximum.\\
a. Prove that $ME^{2} = \frac{3}{2} t^{2} - \frac{5}{2} t + \frac{5}{4}$.\\
b. Prove that triangle $M\mathrm{EG}$ is isosceles at $M$.\\
Deduce that $ME \sin\left(\frac{\alpha}{2}\right) = \frac{1}{2\sqrt{2}}$.\\
c. Justify that $\alpha$ is maximum if and only if $\sin\left(\frac{\alpha}{2}\right)$ is maximum.\\
Deduce that $\alpha$ is maximum if and only if $ME^{2}$ is minimum.\\
d. Conclude.
\end{enumerate}