Problem

Source: Azerbaijan Math Olympiad Training

Tags: geometry, TST



Consider two circles $k_1,k_2$ touching at point $T$. A line touches $k_2$ at point $X$ and intersects $k_1$ at points $A,B$ where $B$ lies between $A$ and $X$.Let $S$ be the second intersection point of $k_1$ with $XT$. On the arc $\overarc{TS}$ not containing $A$ and $B$ , a point $C$ is choosen. Let $CY$ be the tangent line to $k_2$ with $Y\in{k_2}$ , such that the segment $CY$ doesn't intersect the segment $ST$ .If $I=XY\cap{SC}$ , prove that : $(a)$ the points $C,T,Y,I$ are concyclic. $(b)$ $I$ is the $A-excenter$ of $\triangle ABC$