Space is referred to the orthonormal frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the plane $\mathscr{P}$ with equation $2x + y - 2z + 4 = 0$ and the points A with coordinates $(3; 2; 6)$, B with coordinates $(1; 2; 4)$, and C with coordinates $(4; -2; 5)$.

\begin{enumerate}
  \item a. Verify that the points A, B and C define a plane.\\
  b. Verify that this plane is the plane $\mathscr{P}$.
  \item a. Show that the triangle ABC is right-angled.\\
  b. Write a system of parametric equations for the line $\Delta$ passing through O and perpendicular to the plane $\mathscr{P}$.\\
  c. Let K be the orthogonal projection of O onto $\mathscr{P}$. Calculate the distance OK.\\
  d. Calculate the volume of the tetrahedron OABC.
  \item We consider, in this question, the system of weighted points
$$S = \{(\mathrm{O}, 3), (\mathrm{A}, 1), (\mathrm{B}, 1), (\mathrm{C}, 1)\}$$
  a. Verify that this system admits a centroid, which we denote G.\\
  b. Let I denote the centroid of the triangle ABC. Show that G belongs to (OI).\\
  c. Determine the distance from G to the plane $\mathscr{P}$.
  \item Let $\Gamma$ be the set of points $M$ in space satisfying:
$$\|3\overrightarrow{M\mathrm{O}} + \overrightarrow{M\mathrm{A}} + \overrightarrow{M\mathrm{B}} + \overrightarrow{M\mathrm{C}}\| = 5.$$
Determine $\Gamma$. What is the nature of the set of points common to $\mathscr{P}$ and $\Gamma$?
\end{enumerate}