Question 79
Circles $O_1,O_2$ are externally tangent at $T$. A line tangent to $O_2$ at $X$ meets $O_1$ at $A,B$, with $B$ inside segment $AX$. Line $XT$ meets $O_1$ again at $S$. Let $C$ be on the $TS$ not containing $A,B$ (the source's notation is interpreted as the arc). Draw a tangent from $C$ to $O_2$, touching at $Y$, such that segments $CY$ and $ST$ do not intersect. Let $I=SC\cap XY$. Prove:
(1) $C,T,I,Y$ are concyclic;
(2) $I$ is the excenter opposite $A$ of triangle $ABC$.
(1) $C,T,I,Y$ are concyclic;
(2) $I$ is the excenter opposite $A$ of triangle $ABC$.