Problem (2020KoMaLA771): Let $\omega$ denote the incircle of triangle $ABC,$ which is tangent to side $BC$ at point $D.$ Let $G$ denote the second intersection of line $AD$ and circle $\omega.$ The tangent to $\omega$ at point $G$ intersects sides $AB$ and $AC$ at points $E$ and $F$ respectively. The circumscribed circle of $DEF$ intersects $\omega$ at points $D$ and $M.$ The circumscribed circle of $BCG$ intersects $\omega$ at points $G$ and $N.$ Prove that lines $AD$ and $MN$ are parallel.
AlphaGeometry formalization: a b c = triangle; i = incenter a b c; d = foot i b c; g = on_line a d, on_circle i d; e = on_line a b, on_tline g i g; f = on_line a c, on_tline g i g; o1 = circumcenter d e f; m = on_circle i d, on_circle o1 d; o2 = circumcenter b c g; n = on_circle i d, on_circle o2 g ? para a d m n
HAGeo formalization: A B C = triangle; I = incenter A B C; BC = line B C; D = foot I BC; (I) = circle_center_point I D; AD = line A D; G = intersection AD (I); IG = line I G; l1 = perpendicular_line G IG; AB = line A B; E = intersection l1 AB; AC = line A C; F = intersection l1 AC; O1 = circumcenter D E F; (DEF) = circle_center_point O1 D; M = intersection (DEF) (I); O2 = circumcenter B C G; (BCG) = circle_center_point O2 G; N = intersection (BCG) (I); MN = line M N; Prove: parallel AD MN
Difficulty score: 3.4