The image you uploaded presents a geometry problem related to a cube. Here's a summary of the problem:

• You are asked to consider a cube ABCDEFGH.
• Various points are defined in relation to the cube:
  • $M$ is the midpoint of $[BF]$
  • $L$ is the midpoint of $[BC]$
  • $N$ is defined by the relation $\overrightarrow{CN} = \frac{1}{2} \overrightarrow{GC}$
  • $P$ is the center of the face $ADHE$.

The problem is split into two parts:

Partie A

1. Justify that the line $(MN)$ divides segment $[BC]$ in its midpoint $I$.
2. Reproduce the figure and construct the section of the cube by the plane $(MNP)$.

Partie B

The space is defined with the orthonormal reference frame $(A; \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE})$.

1. Justify that the vector $\vec{n} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ is a normal vector to the plane $(MNP)$. Then, deduce the Cartesian equation of the plane $(MNP)$.
2. Determine a parametric representation of the line through $G$ and orthogonal to the plane $(MNP)$.
3. Show that the line $(MN)$ intersects the plane $(MNP)$ at point $K$ with coordinates $(2; -3; 3)$.
4. Calculate the distance $GK$.
5. Prove that the points $M$, $E$, $D$, and $T$ are coplanar, and that the plane formed by them is orthogonal to the line $MEDI$.
6. Calculate the volume of pyramid $GMEDI$.

Would you like help solving a specific part of this problem or additional details on any step?