Let $H$ be the orthocenter of triangle $ABC$. The tangents to the circumcircles of triangles $CHB$ and $AHB$ at point $H$ meet $AC$ at points $A_1$ and $C_1$ respectively. Prove that $A_1H = C_1H$.
Problem
Source: 2011 Sharygin Geometry Olympiad Correspondence Round P9
Tags: geometry, orthocenter, Tangents, equal segments