\textbf{(For candidates who have not followed the specialization course)}

In space equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the tetrahedron ABCD whose vertices have coordinates:
$$\mathrm{A}(1;-\sqrt{3};0);\quad \mathrm{B}(1;\sqrt{3};0);\quad \mathrm{C}(-2;0;0);\quad \mathrm{D}(0;0;2\sqrt{2}).$$

\begin{enumerate}
  \item Prove that the plane (ABD) has the Cartesian equation $4x + z\sqrt{2} = 4$.
  \item We denote by $\mathscr{D}$ the line whose parametric representation is
  $$\left\{\begin{array}{l} x = t \\ y = 0 \\ z = t\sqrt{2} \end{array}, t \in \mathbb{R}\right.$$
  a. Prove that $\mathscr{D}$ is the line that is parallel to $(\mathrm{CD})$ and passes through O.\\
  b. Determine the coordinates of point G, the intersection of the line $\mathscr{D}$ and the plane (ABD).
  \item a. We denote by L the midpoint of segment $[\mathrm{AC}]$.\\
  Prove that the line (BL) passes through point O and is orthogonal to the line (AC).\\
  b. Prove that triangle ABC is equilateral and determine the centre of its circumscribed circle.
  \item Prove that the tetrahedron ABCD is regular, that is, a tetrahedron whose six edges all have the same length.
\end{enumerate}