CONO-SUR 2004 Problem 2

Given a circle $C$ and a point $P$ on its exterior, two tangents to the circle are drawn through $P$ , with $A$ and $B$ being the points of tangency. We take a point $Q$ on the minor arc $AB$ of $C$ . Let $M$ be the intersection of $AQ$ with the line perpendicular to $AQ$ that goes through $P$ , and let $N$ be the intersection of $BQ$ with the line perpendicular to $BQ$ that goes through $P$ . Show that, by varying $Q$ on the minor arc $AB$ , all of the lines $MN$ pass through the same point.