BG-MO 2005 Problem 2

Consider two circles $k_{1},k_{2}$ touching externally at point $T$ . a line touches $k_{2}$ at point $X$ and intersects $k_{1}$ at points $A$ and $B$ . Let $S$ be the second intersection point of $k_{1}$ with the line $XT$ . On the arc $\widehat{TS}$ not containing $A$ and $B$ is chosen a point $C$ . Let $\ CY$ be the tangent line to $k_{2}$ with $Y\in k_{2}$ , such that the segment $CY$ does not intersect the segment $ST$ . If $I=XY\cap SC$ . Prove that :

(a) the points $C,T,Y,I$ are concyclic.

(b) $I$ is the excenter of triangle $ABC$ with respect to the side $BC$ .