Problem (2012G2): Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.
AlphaGeometry formalization: a b c = triangle; d = on_circum a b c; e = on_line a c, on_line b d; f = on_line a d, on_line b c; g = parallelogram d e c g; h = reflect e a d ? cyclic d h f g
HAGeo formalization: A B C = triangle; (ABC) = circle A B C; D = on_circle (ABC); AC = line A C; BD = line B D; E = intersection AC BD; AD = line A D; BC = line B C; F = intersection AD BC; G = parallelogram D E C; H = reflect_point_wrt_line E AD; Prove: concyclic D H F G
Difficulty score: 2.3