Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.

*Proposed by Anant Mudgal, India*

Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$.

Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.

*Proposed by Anant Mudgal, India*

Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.

"Questions we can't answer are far better than answers we can't question"