KMO 2005 Problem 2

For triangle $ABC$ , $P$ and $Q$ satisfy $\angle BPA + \angle AQC=90^{\circ}$ . It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise(or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ( $A \neq N$ , however if $A$ is the only intersection $A=N$ ), and the midpoint of segment $BC$ be $M$ . Show that the length of $MN$ does not depend on $P$ and $Q$ .