Ukrainian Mathematical Olympiad

The segments $BC$ and $AD$ are the bases of a trapezoid $ABCD$. Through the point of intersection of its diagonals, a straight line is drawn parallel to the bases. This line intersects the sides $AB$ and $CD$ at points $M$ and $N$ respectively. The segments $DP$ and $AQ$ are the heights of the triangles $DMC$ and $ABN$ respectively. Prove that $AP=DQ$.