Problem (2023SAGFp2): In an acute-angled triangle, $ABC$ $w$ is a circumscribed circle, and $O$ is inside the triangle and $OB = OC$. The point $D$ is selected on the side of $AB$,so that $OD \parallel BC$. The straight line $AO$ repeatedly intersects the circles $w$ and $(COD)$ at the points $A',P$, respectively. $M$ is the middle of the side of $BC$. Circle $(MAA')$ intersects the line $BC$ at the point $L$. The straight line $AL$ repeatedly intersects the circle $w$ at the point $K$. Prove that $\angle APK = \angle AA'M$.
AlphaGeometry formalization: a b c = triangle; u = circumcenter a b c; o = on_bline b c; d = on_line a b, on_pline o b c; a1 = on_line a o, on_circle u a; o1 = circumcenter c o d; p = on_line a o, on_circle o1 o; m = midpoint b c; o2 = circumcenter m a a1; l = on_line b c, on_circle o2 m; k = on_line a l, on_circle u a ? eqangle a p p k m a1 a1 a
HAGeo formalization: A B C = acute_triangle; U = circumcenter A B C; (ABC) = circle_center_point U A; l = perpendicular_bisector B C; O = on_line l; BC = line B C; l1 = parallel_line O BC; AB = line A B; D = intersection l1 AB; AO = line A O; A1 = intersection AO (ABC); O1 = circumcenter C O D; (CDO) = circle_center_point O1 O; P = intersection AO (CDO); M = midpoint B C; O2 = circumcenter M A A1; (AA1M) = circle_center_point O2 M; L = intersection (AA1M) BC; AL = line A L; K = intersection AL (ABC); Prove: equal_angle A P K M A1 A
Difficulty score: 4.2