Question 45
Two circles $k_{1}$ and $k_{2}$ are externally tangent at point $T$. A line meets $k_{1}$ at points $A$ and $B$ and is tangent to $k_{2}$ at point $X$. The line $X T$ meets $k_{1}$ at point $S$ and let $C$ be a point on the arc $\widehat{T S}$ which does not contain $A$ and $B$. Let $C Y$ be the tangent line to $k_{2}$ ($Y \in k_{2}$) such that the segments $C Y$ and $S T$ do not intersect. If $I$ is the intersection point of the lines $X Y$ and $S C$, prove that:
a) the points $C, T, Y$ and $I$ are concyclic;
b) $I$ is the center of the excircle of $\triangle A B C$ tangent to the side $B C$.
a) the points $C, T, Y$ and $I$ are concyclic;
b) $I$ is the center of the excircle of $\triangle A B C$ tangent to the side $B C$.