Question 128
Given the rectangular cuboid $ABCD A'B'C'D'$, let $M$ be the center of the face $ABB'A'$. Denote by $M_1$ the foot of the perpendicular from $M$ to the line $B'C$ and by $M_2$ the foot of the perpendicular from $M$ to $AD'$. Prove that:
a) $[MM_1] = [MM_2]$;
b) the line $d$ of intersection between the planes $(MM_1M_2)$ and $(ABC)$ is parallel to $AD$;
c) the dihedral angle between the planes $(MM_1M_2)$ and $(ABC)$ is $45^\circ$ if and only if $\frac{BC}{AB} = \frac{BB'}{BC} + \frac{BC}{BB'}$.
a) $[MM_1] = [MM_2]$;
b) the line $d$ of intersection between the planes $(MM_1M_2)$ and $(ABC)$ is parallel to $AD$;
c) the dihedral angle between the planes $(MM_1M_2)$ and $(ABC)$ is $45^\circ$ if and only if $\frac{BC}{AB} = \frac{BB'}{BC} + \frac{BC}{BB'}$.