Problem (2016ARMOg11p8): Medians $AM_A,BM_B,CM_C$ of triangle $ABC$ intersect at $M$.Let $\Omega_A$ be circumcircle of triangle passes through midpoint of $AM$ and tangent to $BC$ at $M_A$.Define $\Omega_B$ and $\Omega_C$ analogusly.Prove that $\Omega_A,\Omega_B$ and $\Omega_C$ intersect at one point.
AlphaGeometry formalization: a b c = triangle; ma mb mc m = centroid a b c; pa = midpoint a m; pb = midpoint b m; pc = midpoint c m; oa = on_tline ma b c, on_bline ma pa; ob = on_tline mb a c, on_bline mb pb; oc = on_tline mc a b, on_bline mc pc; x = on_circle oa ma, on_circle ob mb ? cong x oc oc mc
HAGeo formalization: A B C = triangle; M Ma Mb Mc = centroid A B C; Pa = midpoint A M; Pb = midpoint B M; Pc = midpoint C M; BC = line B C; l1 = perpendicular_line Ma BC; l2 = perpendicular_bisector Ma Pa; Oa = intersection l1 l2; (Oa) = circle_center_point Oa Ma; AC = line A C; l3 = perpendicular_line Mb AC; l4 = perpendicular_bisector Mb Pb; Ob = intersection l3 l4; (Ob) = circle_center_point Ob Mb; AB = line A B; l5 = perpendicular_line Mc AB; l6 = perpendicular_bisector Mc Pc; Oc = intersection l5 l6; X = intersection (Oa) (Ob); Prove: cong X Oc Oc Mc
Difficulty score: 4.6