\textbf{Exercise 3} (5 points)

The purpose of this exercise is to examine, in different cases, whether the altitudes of a tetrahedron are concurrent, that is, to study the existence of an intersection point of its four altitudes.\\
We recall that in a tetrahedron MNPQ, the altitude from M is the line passing through M perpendicular to the plane (NPQ).

\textbf{Part A: Study of particular cases}

We consider a cube ABCDEFGH. We admit that the lines (AG), (BH), (CE) and (DF), called ``main diagonals'' of the cube, are concurrent.

\begin{enumerate}
  \item We consider the tetrahedron ABCE.\\
a. Specify the altitude from E and the altitude from C in this tetrahedron.\\
b. Are the four altitudes of the tetrahedron ABCE concurrent?

  \item We consider the tetrahedron ACHF and work in the coordinate system $(\mathrm{A} ; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.\\
a. Verify that a Cartesian equation of the plane (ACH) is: $x - y + z = 0$.\\
b. Deduce that (FD) is the altitude from F of the tetrahedron ACHF.\\
c. By analogy with the previous result, specify the altitudes of the tetrahedron ACHF from the vertices $\mathrm{A}$, $\mathrm{C}$ and H respectively.\\
Are the four altitudes of the tetrahedron ACHF concurrent?
\end{enumerate}

In the rest of this exercise, a tetrahedron whose four altitudes are concurrent will be called an orthocentric tetrahedron.

\textbf{Part B: A property of orthocentric tetrahedra}

In this part, we consider a tetrahedron MNPQ whose altitudes from vertices M and N intersect at a point K. The lines (MK) and (NK) are therefore perpendicular to the planes (NPQ) and (MPQ) respectively.

\begin{enumerate}
  \item a. Justify that the line (PQ) is perpendicular to the line (MK); we admit likewise that the lines (PQ) and (NK) are perpendicular.\\
b. What can we deduce from the previous question regarding the line (PQ) and the plane (MNK)? Justify the answer.
  \item Show that the edges [MN] and [PQ] are perpendicular.
\end{enumerate}

Thus, we obtain the following property: If a tetrahedron is orthocentric, then its opposite edges are perpendicular in pairs.

\textbf{Part C: Application}

In an orthonormal coordinate system, we consider the points:
$$\mathrm { R } ( - 3 ; 5 ; 2 ) , \mathrm { S } ( 1 ; 4 ; - 2 ) , \mathrm { T } ( 4 ; - 1 ; 5 ) \quad \text { and } \mathrm { U } ( 4 ; 7 ; 3 ) .$$
Is the tetrahedron RSTU orthocentric? Justify.