Problem (2017EGMOp6): Let $ABC$ be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid $G$ and the circumcentre $O$ of $ABC$ in its sides $BC,CA,AB$ are denoted by $G_1,G_2,G_3$ and $O_1,O_2,O_3$, respectively. Show that the circumcircles of triangles $G_1G_2C$, $G_1G_3B$, $G_2G_3A$, $O_1O_2C$, $O_1O_3B$, $O_2O_3A$ and $ABC$ have a common point.
AlphaGeometry formalization: a b c = triangle; x y z g = centroid a b c; o = circumcenter a b c; g1 = reflect g b c; g2 = reflect g a c; g3 = reflect g a b; o1 = reflect o b c; o2 = reflect o a c; o3 = reflect o a b; p = on_circum a b c, on_circum g1 g2 c ? cyclic p o2 o3 a
HAGeo formalization: A B C = acute_triangle; G X Y Z = centroid A B C; O = circumcenter A B C; BC = line B C; G1 = reflect_point_wrt_line G BC; AC = line A C; G2 = reflect_point_wrt_line G AC; AB = line A B; G3 = reflect_point_wrt_line G AB; O1 = reflect_point_wrt_line O BC; O2 = reflect_point_wrt_line O AC; O3 = reflect_point_wrt_line O AB; (CG1G2) = circle C G1 G2; (ABC) = circle A B C; P = intersection (ABC) (CG1G2); Prove: concyclic P O2 O3 A
Difficulty score: 3.5