Problem (2019GOTEEMp1): Let $ABC$ be a scalene triangle. The incircle of $\triangle ABC$ is tangent to sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D, E, F$, respectively. Let $G$ be a point on the incircle of $\triangle ABC$ such that $\angle AGD = 90^\circ$. If lines $DG$ and $EF$ intersect at $P$, prove that $AP$ is parallel to $BC$.
AlphaGeometry formalization: a b c = triangle; i = incenter a b c; d = foot i b c; e = foot i c a; f = foot i a b; o1 = midpoint a d; g = on_circle i d, on_circle o1 a; p = on_line d g, on_line e f ? para a p b c
HAGeo formalization: A B C = triangle; I = incenter A B C; BC = line B C; CA = line C A; AB = line A B; D = foot I BC; E = foot I CA; F = foot I AB; (I) = circle_center_point I D; O1 = midpoint A D; (O1) = circle_center_point O1 A; G = intersection (I) (O1); DG = line D G; EF = line E F; P = intersection DG EF; AP = line A P; Prove: parallel AP BC
Difficulty score: 2.5