Problem (2019G1): Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
AlphaGeometry formalization: a b c = triangle; o = free; d = on_line a b, on_circle o a; e = on_line a c, on_circle o a; f = on_line b c, on_circle o a; g = on_line b c, on_circle o a; o1 = circumcenter b d f; o2 = circumcenter c e g; t = on_tline f o1 f, on_tline g o2 g ? para a t b c
HAGeo formalization: A B C = triangle; O = point; AB = line A B; Γ = circle_center_point O A; D = intersection Γ AB; AC = line A C; E = intersection Γ AC; BC = line B C; F = intersection BC Γ; G = intersection BC Γ; (BDF) = circle B D F; l1 = tangent F (BDF); (CEG) = circle C E G; l2 = tangent G (CEG); T = intersection l1 l2; AT = line A T; Prove: parallel AT BC
Difficulty score: 1.2