IR-TST 2020 Problem 9

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

Proposed by Alireza Dadgarnia and Amir Parsa Hosseini