Problem (2016USATSTSTp2): Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $OM$ at $P$. Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$.
AlphaGeometry formalization: a b c = triangle; h = orthocenter a b c; o = circumcenter a b c; m = midpoint a h; n = midpoint b c; g = on_circle m a, on_circle o a; q = on_line a n, on_circle m a; p = on_tline g m g, on_line o m; o1 = circumcenter g n q; o2 = circumcenter m b c; t = on_circle o1 g, on_circle o2 m ? coll t p n
HAGeo formalization: A B C = triangle; H = orthocenter A B C; O = circumcenter A B C; M = midpoint A H; N = midpoint B C; (M) = circle_center_point M A; (O) = circle_center_point O A; G = intersection (M) (O); AN = line A N; Q = intersection (M) AN; MG = line M G; l1 = perpendicular_line G MG; MO = line M O; P = intersection l1 MO; O1 = circumcenter G N Q; (O1) = circle_center_point O1 G; O2 = circumcenter B C M; (O2) = circle_center_point O2 M; T = intersection (O1) (O2); Prove: collinear P N T
Difficulty score: 3.7