SHARYGIN 2016 Problem 9

Let $ABC$ be a right-angled triangle and $CH$ be the altitude from its right angle $C$ . Points $O_1$ and $O_2$ are the incenters of triangles $ACH$ and $BCH$ respectively, $P_1$ and $P_2$ are the touching points of their incircles with $AC$ and $BC$ . Prove that lines $O_1P_1$ and $O_2P_2$ meet on $AB$ .