Problem (2021GOWACAp1): In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $P$ be the reflection of $A$ over $\overline{OI}$ and $Q$ be the second intersection of line $\overline{OI}$ with $(BIC)$. Prove that lines $\overline{AI}$, $\overline{BC}$, $\overline{PQ}$ concur.
AlphaGeometry formalization: a b c = triangle; i = incenter a b c; o = circumcenter a b c; p = reflect a o i; o1 = circumcenter b i c; q = on_line o i, on_circle o1 b; r = on_line a i, on_line b c ? coll p q r
HAGeo formalization: A B C = triangle; I = incenter A B C; O = circumcenter A B C; IO = line I O; P = reflect_point_wrt_line A IO; O1 = circumcenter B C I; (O1) = circle_center_point O1 B; Q = intersection IO (O1); AI = line A I; BC = line B C; R = intersection AI BC; Prove: collinear P Q R
Difficulty score: 2.3