Question 105
Let $ABCD A'B'C'D'$ be a cuboid so that $ABB'A'$ is a square, and let the points $M \in (AC)$, $N \in (BC')$ be such that $AM = BN$ and $MN \perp BD'$.
a) Prove that $M$ and $N$ are the midpoints of the segments $AC$, respectively $BC'$.
b) If, moreover, $d(MN, BD') = \frac{1}{6}AN$ and $AB \ge BC$, then $ABCD A'B'C'D'$ is a cube.
a) Prove that $M$ and $N$ are the midpoints of the segments $AC$, respectively $BC'$.
b) If, moreover, $d(MN, BD') = \frac{1}{6}AN$ and $AB \ge BC$, then $ABCD A'B'C'D'$ is a cube.