MathNet 2020 — Math Olympiad Vault — Part 05 Question 112

Let $ABC$ be a triangle with a right angle at $C$, let $I$ be its incenter, and let $H$ be the orthogonal projection of $C$ on $AB$. The incircle $\omega$ of the triangle $ABC$ is tangent to the sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in the lines $C_1A_1$ and $C_1B_1$, respectively, and let $K$ and $L$ be the reflections of $H$ in the same lines. Prove that the circles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.
Russia, Dmitry Prokopenko