Question 172
Let $ABC$ be a triangle that is not isosceles at $A$. Denote by $D, E, F$ the points of tangency of the incircle with the sides $(BC)$, $(AC)$ and $(AB)$, and by $I$ the incentre of $ABC$. Let $P$ and $Q$ be the intersections of the line $(EF)$ with the circle $\Omega$ circumscribed about $ABC$. Finally, let $O_{1}$ and $O_{2}$ be the centres of the circumcircles of $AIB$ and $AIC$. Show that the centre of the circumcircle of $DPQ$ lies on the line $\left(O_{1}O_{2}\right)$.