OBM 2014 Problem 6

Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$ . Circle $\omega_A$ is externally tangent to $\omega$ and tangent to sides $AB$ and $AC$ at $A_1$ and $A_2$ , respectively. Let $r_A$ be the line $A_1A_2$ . Define $r_B$ and $r_C$ in a similar fashion. Lines $r_A$ , $r_B$ and $r_C$ determine a triangle $XYZ$ . Prove that the incenter of $XYZ$ , the circumcenter of $XYZ$ and $I$ are collinear.