Question 113
In triangle $ABC$ let $I_a$ be the $A$-excenter. Let $\omega$ be an arbitrary circle that passes through $A$ and $I_a$ and intersects the extensions of sides $AB$ and $AC$ (extended from $B$ and $C$) at $X$ and $Y$, respectively. Let $S$ and $T$ be points on segments $I_aB$ and $I_aC$, respectively, such that $\widehat{AXI_a} = \widehat{BTI_a}$ and $\widehat{AYI_a} = \widehat{CSI_a}$. Suppose that lines $BT$ and $CS$ intersect at $K$, and lines $KI_a$ and $TS$ intersect at $Z$. Prove that $X, Y, Z$ are collinear.