\textbf{Exercise 4 (7 points) -- Geometry in the plane and in space}

Consider the cube ABCDEFGH. Let I be the midpoint of segment [EH] and consider the triangle CFI. The space is equipped with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$ and we admit that point I has coordinates $\left(0; \frac{1}{2}; 1\right)$ in this coordinate system.

\begin{enumerate}
  \item a. Give without justification the coordinates of points C, F and G.\\
  b. Prove that the vector $\vec{n}\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ is normal to the plane (CFI).\\
  c. Verify that a Cartesian equation of the plane (CFI) is: $x + 2y + 2z - 3 = 0$.
  \item Let $d$ be the line passing through G and perpendicular to the plane (CFI).\\
  a. Determine a parametric representation of the line $d$.\\
  b. Prove that the point $\mathrm{K}\left(\frac{7}{9}; \frac{5}{9}; \frac{5}{9}\right)$ is the orthogonal projection of point G onto the plane (CFI).\\
  c. Deduce from the previous questions that the distance from point G to the plane (CFI) is equal to $\frac{2}{3}$.
  \item Consider the pyramid GCFI.\\
  Recall that the volume $V$ of a pyramid is given by the formula
$$V = \frac{1}{3} \times b \times h$$
where $b$ is the area of a base and $h$ is the height associated with this base.\\
  a. Prove that the volume of the pyramid GCFI is equal to $\frac{1}{6}$, expressed in cubic units.\\
  b. Deduce the area of triangle CFI, in square units.
\end{enumerate}