Question 42
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 $IG$.