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there are two circles intersecting each at two points

Source: VietNam TST 2004, problem 3

May 9, 2004
geometrygeometric transformationrotationhomothetyfunctionratiogeometry solved

Problem Statement

In the plane, there are two circles Γ1,Γ2\Gamma_1, \Gamma_2 intersecting each other at two points AA and BB. Tangents of Γ1\Gamma_1 at AA and BB meet each other at KK. Let us consider an arbitrary point MM (which is different of AA and BB) on Γ1\Gamma_1. The line MAMA meets Γ2\Gamma_2 again at PP. The line MKMK meets Γ1\Gamma_1 again at CC. The line CACA meets Γ2\Gamma_2 again at QQ. Show that the midpoint of PQPQ lies on the line MCMC and the line PQPQ passes through a fixed point when MM moves on Γ1\Gamma_1. [Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .]
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