Question 91
Let $I$ be the incenter of a triangle $ABC$ whose incircle is tangent to the sides $BC$, $AC$, $AB$ at $D$, $E$, $F$, respectively. Suppose the circumcircle of the triangle $ABC$ intersects the line $EF$ at $P$ and $Q$. If $O_1$ and $O_2$ are the circumcenters of the triangles $IAB$ and $IAC$, respectively, show that the circumcenter of the triangle $DPQ$ lies on the line $O_1O_2$.