USEMO 2020 Problem 3

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$ . Let $\Gamma$ denote the circumcircle of triangle $ABC$ , and $N$ the midpoint of $OH$ . The tangents to $\Gamma$ at $B$ and $C$ , and the line through $H$ perpendicular to line $AN$ , determine a triangle whose circumcircle we denote by $\omega_A$ . Define $\omega_B$ and $\omega_C$ similarly. Prove that the common chords of $\omega_A$ , $\omega_B$ and $\omega_C$ are concurrent on line $OH$ .

Proposed by Anant Mudgal