SHARYGIN 2018 Problem 6

Let $CH$ be the altitude of a right-angled triangle $ABC$ ($\angle C = 90^\circ$) with $BC = 2AC$. Let $O_1, O_2$ and $O$ be the incenters of triangles $ACH, BCH$ and $ABC$ respectively, and $H_1, H_2, H_0$ be the projections of $O_1, O_2, O$ respectively to $AB$. Prove that $H_1H = HH_0 = H_0H_2$.