PE-MO 2011 Problem 3

Let $ABC$ be a right triangle, right in $B$ . Inner bisectors are drawn $CM$ and $AN$ that intersect in $I$ . Then, the $AMIP$ and $CNIQ$ parallelograms are constructed. Let $U$ and $V$ are the midpoints of the segments $AC$ and $PQ$ , respectively. Prove that $UV$ is perpendicular to $AC$ .