PMO 2022 Problem 4

Let $\triangle ABC$ have incenter $I$ and centroid $G$ . Suppose that $P_A$ is the foot of the perpendicular from $C$ to the exterior angle bisector of $B$ , and $Q_A$ is the foot of the perpendicular from $B$ to the exterior angle bisector of $C$ . Define $P_B$ , $P_C$ , $Q_B$ , and $Q_C$ similarly. Show that $P_A, P_B, P_C, Q_A, Q_B,$ and $Q_C$ lie on a circle whose center is on line $IG$ .