Putnam 1963 A6

Let $U$ and $V$ be any two distinct points on an ellipse, let $M$ be the midpoint of the chord $UV$ , and let $AB$ and $CD$ be any two other chords through $M$ . If the line $UV$ meets the line $AC$ in the point $P$ and the line $BD$ in the point $Q$ , prove that $M$ is the midpoint of the segment $PQ.$