BY-MO 2007 Problem 5

Let $O$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$ , $AO = CO$ . Points $P$ and $Q$ are marked on the segments $AO$ and $CO$ , respectively, such that $PO = OQ$ . Let $N$ and $K$ be the intersection points of the sides $AB$ , $CD$ , and the lines $DP$ and $BQ$ respectively. Prove that the points $N$ , $O$ , and $K$ are colinear.