Question 21
Let the line segment $[AB]$ be a chord of the circle $\Gamma$ not passing through the center of it and $M$ be the midpoint of $[AB]$. Let $C$ be a variable point on $\Gamma$ different from $A$ and $B$, and let $P$ be the point where the tangent line to the circumcircle of the triangle $CAM$ at the point $A$ meets the tangent line to the circumcircle of the triangle $CBM$ at the point $B$. Show that the lines $CP$ pass through a common fixed point as $C$ varies.