Given a circle $k$ and the point $P$ outside it, an arbitrary line $s$ passing through $P$ intersects $k$ at the points $A$ and $B$ . Let $M$ and $N$ be the midpoints of the arcs determined by the points $A$ and $B$ and let $C$ be the point on $AB$ such that $PC^2=PA\cdot PB$ . Prove that $\angle MCN$ doesn't depend on the choice of $s$. [Moderator edit: This problem has also been discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=56295 .]