Problem

Source: Kyiv City MO 2022 Round 2, Problem 10.3

Tags: geometry, ratio



Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Prove that if $\frac{H_BC}{AC} = \frac{H_CA}{AB}$, then the line symmetric to $BC$ with respect to line $H_BH_C$ is tangent to the circumscribed circle of triangle $H_BH_CA$. (Proposed by Mykhailo Bondarenko)