Question 23
Let $\triangle ABC$ be a triangle with a right angle at $C$ and $|AC| \gt |BC|$, let $I$ be the centre of the incircle and let $H$ be the projection of $C$ onto the segment $AB$. The incircle $\omega$ of $\triangle ABC$ touches the sides $BC$, $CA$ and $AB$ at the points $A_{1}$, $B_{1}$ and $C_{1}$, respectively. Let $E$ and $F$ be the reflections of $C$ in the lines $A_{1}C_{1}$ and $B_{1}C_{1}$, respectively, and let $K$ and $L$ be the reflections of $H$ in the lines $A_{1}C_{1}$ and $B_{1}C_{1}$, respectively. Prove that the circumcircles of $A_{1}EI$, $B_{1}FI$ and $C_{1}KL$ pass through a common point.