Question 89
Given a triangle $\triangle ABC$ with circumcircle $\Gamma$. Points $E$ and $F$ are the feet of angle bisectors of $B$ and $C$, let $I$ be incenter and $K$ be the intersection point of $AI$ and $EF$. Suppose that $T$ is the midpoint of arc $\widehat{BAC}$. Circle $\Gamma$ intersects the $A$-median and circumcircle of $\triangle AEF$ for the second time at $X$ and $S$. Let $S'$ be the reflection of $S$ with respect to $AI$ and $J$ be the second intersection point of circumcircle of $\triangle AS'K$ and $AX$. Prove that quadrilateral $TJIX$ is cyclic.