We place ourselves in an orthonormal coordinate system with origin O and axes $( \mathrm { O } x )$, $( \mathrm { O } y )$ and $( \mathrm { O } z )$.\\
In this coordinate system, we are given the points $\mathrm { A } ( - 3 ; 0 ; 0 ) , \mathrm { B } ( 3 ; 0 ; 0 ) , \mathrm { C } ( 0 ; 3 \sqrt { 3 } ; 0 )$ and $\mathrm { D } ( 0 ; \sqrt { 3 } ; 2 \sqrt { 6 } )$.\\
We denote H as the midpoint of segment [CD] and I as the midpoint of segment [BC].

\begin{enumerate}
  \item Calculate the lengths AB and AD.
\end{enumerate}

We admit for the rest that all edges of the solid ABCD have the same length, that is, the tetrahedron ABCD is a regular tetrahedron.\\
We call $\mathscr { P }$ the plane with normal vector $\overrightarrow { \mathrm { OH } }$ and passing through point I.\\
2. Study of the cross-section of tetrahedron ABCD by plane $\mathscr { P }$\\
a. Show that a Cartesian equation of plane $\mathscr { P }$ is : $2 y \sqrt { 3 } + z \sqrt { 6 } - 9 = 0$.\\
b. Prove that the midpoint J of $[ \mathrm { BD } ]$ is the intersection point of line (BD) and plane $\mathscr { P }$.\\
c. Give a parametric representation of line (AD), then prove that plane $\mathscr { P }$ and line (AD) intersect at a point K whose coordinates you will determine.\\
d. Prove that lines (IJ) and (JK) are perpendicular.\\
e. Determine precisely the nature of the cross-section of tetrahedron ABCD by plane $\mathscr { P }$.\\
3. Can we place a point M on edge $[ \mathrm { BD } ]$ such that triangle OIM is right-angled at M?