Problem

Source: Mexico National Olympiad 2021Problem 4

Tags: geometry, tangent circles, orthocenter, Euler Line, circumcircle



Let $ABC$ be an acutangle scalene triangle with $\angle BAC = 60^{\circ}$ and orthocenter $H$. Let $\omega_b$ be the circumference passing through $H$ and tangent to $AB$ at $B$, and $\omega_c$ the circumference passing through $H$ and tangent to $AC$ at $C$. Prove that $\omega_b$ and $\omega_c$ only have $H$ as common point. Prove that the line passing through $H$ and the circumcenter $O$ of triangle $ABC$ is a common tangent to $\omega_b$ and $\omega_c$. Note: The orthocenter of a triangle is the intersection point of the three altitudes, whereas the circumcenter of a triangle is the center of the circumference passing through it's three vertices.