RMM 2020 Problem 1

Let $ABC$ be a triangle with a right angle at $C$ . Let $I$ be the incentre of triangle $ABC$ , and let $D$ be the foot of the altitude from $C$ to $AB$ . The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$ , $CA$ , and $AB$ at $A_1$ , $B_1$ , and $C_1$ , respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$ , respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$ , respectively.

Prove that the circumcircles of triangles $A_1EI$ , $B_1FI$ , and $C_1KL$ have a common point.