Question 49
Let $ABC$ be a scalene acute-angled triangle, $\Gamma$ its circumcircle, and $K$ the foot of the bisector from vertex $A$. Let $M$ be the midpoint of the arc $BC$ containing $A$. Let $A'$ be the second intersection of $MK$ with $\Gamma$, and let $T$ be the intersection of the tangents to $\Gamma$ at $A$ and at $A'$. Moreover, let $R$ be the intersection of the perpendicular to $AK$ through $A$ and the perpendicular to $A'K$ through $A'$. Prove that $T$, $R$ and $K$ are collinear.