IR-TST 2022 Problem 3

Incircle $\omega$ of triangle $ABC$ is tangent to sides $CB$ and $CA$ at $D$ and $E$ , respectively. Point $X$ is the reflection of $D$ with respect to $B$ . Suppose that the line $DE$ is tangent to the $A$ -excircle at $Z$ . Let the circumcircle of triangle $XZE$ intersect $\omega$ for the second time at $K$ . Prove that the intersection of $BK$ and $AZ$ lies on $\omega$ .

Proposed by Mahdi Etesamifard and Alireza Dadgarnia