Problem (2020USATSTSTp2b): Let $ABC$ be a scalene triangle with incenter $I$. The incircle of $ABC$ touches $\overline{BC},\overline{CA},\overline{AB}$ at points $D,E,F$, respectively. Let $P$ be the foot of the altitude from $D$ to $\overline{EF}$, and let $M$ be the midpoint of $\overline{BC}$. The rays $AP$ and $IP$ intersect the circumcircle of triangle $ABC$ again at points $G$ and $Q$, respectively. Show that the incenter of triangle $GQM$ coincides with $D$.
AlphaGeometry formalization: a b c = triangle; i = incenter a b c; d = foot i b c; e = foot i a c; f = foot i a b; p = foot d e f; m = midpoint b c; g = on_line a p, on_circum a b c; q = on_line i p, on_circum a b c ? eqangle q m m d m d m g
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; EF = line E F; P = foot D EF; M = midpoint B C; AP = line A P; (ABC) = circle A B C; G = intersection AP (ABC); IP = line I P; Q = intersection IP (ABC); Prove: equal_angle Q M D D M G
Difficulty score: 3.1