Problem (2008CHNSouthEastMOp3): In $\triangle ABC$, side $BC>AB$. Point $D$ lies on side $AC$ such that $\angle ABD=\angle CBD$. Points $Q,P$ lie on line $BD$ such that $AQ\bot BD$ and $CP\bot BD$. $M,E$ are the midpoints of side $AC$ and $BC$ respectively. Circle $O$ is the circumcircle of $\triangle PQM$ intersecting side $AC$ at $H$. Prove that $O,H,E,M$ lie on a circle
AlphaGeometry formalization: a b c = triangle; d = on_line a c, angle_bisector a b c; q = foot a b d; p = foot c b d; m = midpoint a c; e = midpoint b c; o = circumcenter p q m; h = on_line a c, on_circle o p ? cyclic o h e m
HAGeo formalization: A B C = triangle; l1 = angle_bisector A B C; AC = line A C; D = intersection l1 AC; BD = line B D; Q = foot A BD; P = foot C BD; M = midpoint A C; E = midpoint B C; O = circumcenter P Q M; (O) = circle_center_point O P; H = intersection AC (O); Prove: concyclic O H E M
Difficulty score: 2.3