ELMO 2010 Problem 6

Let $ABC$ be a triangle with circumcircle $\omega$ , incenter $I$ , and $A$ -excenter $I_A$ . Let the incircle and the $A$ -excircle hit $BC$ at $D$ and $E$ , respectively, and let $M$ be the midpoint of arc $BC$ without $A$ . Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$ . If $TI$ intersects $\omega$ again at $S$ , prove that $SI_A$ and $ME$ meet on $\omega$ .

Amol Aggarwal.