Problem (2018CGMOp2): Points $D,E$ lie on segments $AB,AC$ of $\triangle ABC$ such that $DE\parallel BC$. Let $O_1,O_2$ be the circumcenters of $\triangle ABE, \triangle ACD$ respectively. Line $O_1O _2$ meets $AC$ at $P$, and $AB$ at $Q$. Let $O$ be the circumcenter of $\triangle APQ$, and $M$ be the intersection of $AO$ extended and $BC$. Prove that $M$ is the midpoint of $BC$.
AlphaGeometry formalization: a b c = triangle; d = on_line a b; e = on_line a c, on_pline d b c; o1 = circumcenter a b e; o2 = circumcenter a c d; p = on_line o1 o2, on_line a c; q = on_line o1 o2, on_line a b; o3 = circumcenter a p q; m = on_line a o3, on_line b c ? midp m b c
HAGeo formalization: A B C = triangle; AB = line A B; D = on_line AB; BC = line B C; l1 = parallel_line D BC; AC = line A C; E = intersection AC l1; O1 = circumcenter A B E; O2 = circumcenter A C D; O1O2 = line O1 O2; P = intersection O1O2 AC; Q = intersection O1O2 AB; O3 = circumcenter A P Q; AO3 = line A O3; M = intersection AO3 BC; Prove: midpoint M B C
Difficulty score: 2.7