We consider the cube ABCDEFGH below such that $\mathrm { AB } = 1$.\\
We denote M the center of face BCGF and N the center of face EFGH.

We use the orthonormal coordinate system ( D ; $\overrightarrow { \mathrm { DH } } , \overrightarrow { \mathrm { DC } } , \overrightarrow { \mathrm { DA } }$ ).

\begin{enumerate}
  \item Give without justification the coordinates of points F and C.
  \item Calculate the coordinates of points M and N.
  \item a. Prove that the vector $\overrightarrow { \mathrm { AG } }$ is normal to the plane (HFC).\\
b. Deduce a Cartesian equation of the plane (HFC).
  \item Determine a parametric representation of the line (AG).
  \item Prove that the point R with coordinates $\left( \frac { 2 } { 3 } ; \frac { 2 } { 3 } ; \frac { 1 } { 3 } \right)$ is the orthogonal projection of point G onto the plane (HFC).
  \item We admit that a parametric representation of the line (FG) is:
$$\left\{ \begin{array} { l } 
x = 1 \\
y = 1 \quad ( t \in \mathbb { R } ) . \\
z = t
\end{array} \right.$$
Prove that there exists a unique point K on the line (FG) such that the triangle KMN is right-angled at K.
  \item What fraction of the volume of cube ABCDEFGH does the volume of tetrahedron FNKM represent?
\end{enumerate}