Question 151
In $\triangle ABC$, let $M$ be the midpoint of side $BC$. Line $AM$ meets the circumcircle $\Gamma$ of $\triangle ABC$ again at $R$, and the line through $R$ parallel to $BC$ meets $\Gamma$ again at $S$. Let $U$ be the foot of the perpendicular from $R$ to $BC$, and let $T$ be the reflection of $U$ in $R$. Let $D$ be the point on $BC$ such that $AD$ is the altitude of $\triangle ABC$, and let $N$ be the midpoint of $AD$. Finally, let $AS$ and $MN$ meet at $K$. Prove that $AT$ bisects $MK$.