Problem (2022KoMaLA805): In acute triangle $ABC,$ the feet of the altitudes are $A_1,B_1,$ and $C_1$ (with the usual notations on sides $BC,CA,$ and $AB$ respectively). The circumcircles of triangles $AB_1C_1$ and $BC_1A_1$ intersect at the circumcircle of triangle $ABC$ ar points $P\neq A$ and $Q\neq B,$ respectively. Prove that lines $AQ, BP$ and the Euler line of triangle $ABC$ are either concurrent or parallel to each other.
AlphaGeometry formalization: a b c = triangle; a1 = foot a b c; b1 = foot b a c; c1 = foot c a b; o = circumcenter a b c; o1 = circumcenter a b1 c1; p = on_circle o a, on_circle o1 a; o2 = circumcenter b c1 a1; q = on_circle o b, on_circle o2 b; h = orthocenter a b c; r = on_line a q, on_line b p ? coll r o h
HAGeo formalization: A B C = acute_triangle; BC = line B C; A1 = foot A BC; AC = line A C; B1 = foot B AC; AB = line A B; C1 = foot C AB; O = circumcenter A B C; O1 = circumcenter A B1 C1; (O) = circle_center_point O A; (O1) = circle_center_point O1 A; P = intersection (O1) (O); O2 = circumcenter B C1 A1; (O2) = circle_center_point O2 B; Q = intersection (O2) (O); H = orthocenter A B C; AQ = line A Q; BP = line B P; R = intersection AQ BP; Prove: collinear R O H
Difficulty score: 1.9