Question 127
Let $ABC$ be an acute triangle with $AB \lt AC$. Let $I$ be its incenter and $\Gamma$ be its circumcircle. Let $M$ be the midpoint of $BC$, $K$ the midpoint of arc $BC$ not containing $A$, $L$ the midpoint of arc $BC$ containing $A$ and $J$ the reflection of $I$ with respect to the line $KL$. The line $LJ$ cuts $\Gamma$ again at $T$. The line $TM$ cuts $\Gamma$ again at $S$. Prove that the points $S$, $I$, $M$ and $K$ are concyclic.