Question 6
Let $ABC$ be an acute triangle with $AB\ne AC$, incenter $I$ and incircle $\omega$. Let $D$, $E$, $F$ denote the tangency points of $\omega$ with $\overline{BC}$, $\overline{CA}$, $\overline{AB}$. The line through $D$ perpendicular to $\overline{EF}$ meets $\omega$ again at $R$ (other than $D$), and line $AR$ meets $\omega$ again at $P$ (other than $R$). Suppose the circumcircles of $\triangle PCE$ and $\triangle PBF$ meet again at $Q$ (other than $P$). Prove that lines $DI$ and $PQ$ meet on the external $\angle A$-bisector. The external $\angle A$-bisector is the line through $A$ perpendicular to $AI$.