Exercise 4 (Candidates who have not followed the specialization course)
We connect the centres of each face of a cube ABCDEFGH to form a solid IJKLMN. More precisely, the points $\mathrm { I } , \mathrm { J } , \mathrm { K } , \mathrm { L } , \mathrm { M }$ and N are the centres respectively of the square faces ABCD, BCGF, CDHG, ADHE, ABFE and EFGH (thus the midpoints of the diagonals of these squares).

1. Without using a coordinate system (and thus coordinates) in the reasoning, justify that the lines (IN) and (ML) are orthogonal.

In what follows, we consider the orthonormal coordinate system $( \mathrm { A } ; \overrightarrow { \mathrm { AB } } ; \overrightarrow { \mathrm { AD } } ; \overrightarrow { \mathrm { AE } } )$ in which, for example, the point N has coordinates $\left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; 1 \right)$.

1. a. Give the coordinates of the vectors $\overrightarrow { \mathrm { NC } }$ and $\overrightarrow { \mathrm { ML } }$. b. Deduce that the lines (NC) and (ML) are orthogonal. c. From the previous questions, deduce a Cartesian equation of the plane (NCI).
2. a. Show that a Cartesian equation of the plane (NJM) is: $x - y + z = 1$. b. Is the line (DF) perpendicular to the plane (NJM)? Justify. c. Show that the intersection of the planes (NJM) and (NCI) is a line for which you will give a point and a direction vector. Name the line thus obtained using two points from the figure.

16. As shown in the figure, in the right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$, given $A C \perp B C$. Let D be the midpoint of $A B _ { 1 }$, $B _ { 1 } C \cap B C _ { 1 } = E$ . Prove: (1) $D E \parallel$ plane $A A _ { 1 } C C _ { 1 }$
(2) $B C _ { 1 } \perp A B _ { 1 }$ [Figure]