IT-CESENATICO 2009 Problem 5

Let $ABC$ be an acute-angled scalene triangle and $\Gamma$ be its circumcircle. $K$ is the foot of the internal bisector of $\angle BAC$ on $BC$ . Let $M$ be the midpoint of the arc $BC$ containing $A$ . $MK$ intersect $\Gamma$ again at $A'$ . $T$ is the intersection of the tangents at $A$ and $A'$ . $R$ is the intersection of the perpendicular to $AK$ at $A$ and perpendicular to $A'K$ at $A'$ . Show that $T, R$ and $K$ are collinear.