OMM 2021 Problem 4

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.