Problem (2024BalkanMOp1): Let $ABC$ be an acute-angled triangle with $AC > AB$ and let $D$ be the foot of the $A$-angle bisector on $BC$. The reflections of lines $AB$ and $AC$ in line $BC$ meet $AC$ and $AB$ at points $E$ and $F$ respectively. A line through $D$ meets $AC$ and $AB$ at $G$ and $H$ respectively such that $G$ lies strictly between $A$ and $C$ while $H$ lies strictly between $B$ and $F$. Prove that the circumcircles of $\triangle EDG$ and $\triangle FDH$ are tangent to each other.
AlphaGeometry formalization: a b c = triangle; d = on_line b c, angle_bisector b a c; a1 = reflect a b c; e = on_line a c, on_line a1 b; f = on_line a b, on_line a1 c; g = on_line a c; h = on_line a b, on_line d g; o1 = circumcenter e d g; o2 = circumcenter f d h ? coll o1 o2 d
HAGeo formalization: A B C = acute_triangle; l1 = angle_bisector C A B; BC = line B C; D = intersection l1 BC; AB = line A B; A1 = reflect_point_wrt_line A BC; A1B = line A1 B; AC = line A C; E = intersection A1B AC; A1C = line A1 C; F = intersection AB A1C; l = line_through D; G = intersection l AC; H = intersection l AB; O1 = circumcenter D E G; O2 = circumcenter D F H; Prove: collinear O1 O2 D
Difficulty score: 2.0