MathNet 2023-part-05 — 191 problems (EN–VI) Question 80

Let $M$ and $N$ be the centres of the faces $A'B'C'D'$, respectively $ADD'A'$ of a rectangular cuboid $ABCDA'B'C'D'$. It is known that $AM \perp A'C$ and $C'N \perp BD'$. Prove that $ABCDA'B'C'D'$ is a cube.