Question 51
Consider a cube $ABCD A'B'C'D'$ and two points $M \in (CD')$ and $N \in (DA')$. Prove that $MN$ is the common perpendicular of the lines $CD'$ and $DA'$ if and only if
$$
\frac{D'M}{D'C} = \frac{DN}{DA'} = \frac{1}{3}.
$$
$$
\frac{D'M}{D'C} = \frac{DN}{DA'} = \frac{1}{3}.
$$