ELMO SL 2017 G4

Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\omega$ . Suppose a circle $\omega_B$ is tangent to $BA,BC$ , and internally tangent to $\omega$ at $B_1$ , while a circle $\omega_C$ is tangent to $CA, CB$ , and internally tangent to $\omega$ at $C_1$ . If $B_2, C_2$ are the points opposite to $B,C$ on $\omega$ , respectively, and $X$ denotes the intersection of $B_1C_2, B_2C_1$ , prove that $XA=XI$ .

[i]Proposed by Vincent Huang and Nathan Weckwerth