Problem (2022CHNSouthEastMOg11p6): Let $O$ be the circumcenter of $\triangle ABC$. A circle with center $P$ pass through $A$ and $O$ and $OP$//$BC$. $D$ is a point such that $\angle DBA = \angle DCA = \angle BAC$. Prove that: Circle $(P)$, circle $(BCD)$ and the circle with diameter $(AD)$ share a common point.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; p = on_bline a o, on_pline o b c; d = on_aline d b a b a c, on_aline d c a c a b; q = midpoint a d; k = on_circle p a, on_circle q a ? cyclic k b c d
HAGeo formalization: A B C = triangle; O = circumcenter A B C; l1 = perpendicular_bisector A O; BC = line B C; l2 = parallel_line O BC; P = intersection l1 l2; l3 = angle_equal1 B A B A C; l4 = angle_equal1 C A C A B; D = intersection l3 l4; Q = midpoint A D; (P) = circle_center_point P A; (Q) = circle_center_point Q A; K = intersection (P) (Q); Prove: concyclic K B C D
Difficulty score: 4.0