VN-TST 2004 Problem 3

In the plane, there are two circles $\Gamma_1, \Gamma_2$ intersecting each other at two points $A$ and $B$ . Tangents of $\Gamma_1$ at $A$ and $B$ meet each other at $K$ . Let us consider an arbitrary point $M$ (which is different of $A$ and $B$ ) on $\Gamma_1$ . The line $MA$ meets $\Gamma_2$ again at $P$ . The line $MK$ meets $\Gamma_1$ again at $C$ . The line $CA$ meets $\Gamma_2$ again at $Q$ . Show that the midpoint of $PQ$ lies on the line $MC$ and the line $PQ$ passes through a fixed point when $M$ moves on $\Gamma_1$ .

[Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .]