Question 3
Let $ABC$ be a scalene triangle with incenter $I$. The incircle of $ABC$ touches $BC$, $CA$, $AB$ at points $D$, $E$, $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $EF$, and let $M$ be the midpoint of $BC$. 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 $D$.