Question 145
A point $M$ lies inside a cube $ABCD-A_1B_1C_1D_1$. Points $A', B', C', D', A'_1, B'_1, C'_1$ and $D'_1$ belong to the half-lines $MA, MB, MC, MD, MA_1, MB_1, MC_1$ and $MD_1$ respectively. Prove that if the polyhedron $A'B'C'D'A'_1B'_1C'_1D'_1$ is a parallelepiped (i.e., all its faces are parallelograms), then in fact it is a cube.