IR-TST 2017 Problem 3

In triangle $ABC$ let $I_a$ be the $A$ -excenter. Let $\omega$ be an arbitrary circle that passes through $A,I_a$ and intersects the extensions of sides $AB,AC$ (extended from $B,C$ ) at $X,Y$ respectively. Let $S,T$ be points on segments $I_aB,I_aC$ respectively such that $\angle AXI_a=\angle BTI_a$ and $\angle AYI_a=\angle CSI_a$ .Lines $BT,CS$ intersect at $K$ . Lines $KI_a,TS$ intersect at $Z$ . Prove that $X,Y,Z$ are collinear.

Proposed by Hooman Fattahi