CZ-SK-MO 1957 Problem 4

Consider a non-zero convex angle $\angle POQ$ and its inner point $M$ . Moreover, let $m>0$ be given. Construct a trapezoid $ABCD$ satisfying the following conditions: (1) vertices $A, D$ lie on ray $OP$ and vertices $B,C$ lie on ray $OQ$ , (2) diagonals $AC$ and $BD$ intersect in $M$ , (3) $AB=m$ . Prove that your construction is correct and discuss conditions of solvability.