HMMT 2021 T9

Let scalene triangle $A B C$ have circumcenter $O$ and incenter $I$. Its incircle $\omega$ is tangent to sides $B C, C A$, and $A B$ at $D, E$, and $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $E F$, and let line $D P$ intersect $\omega$ again at $Q \neq D$. The line $O I$ intersects the altitude from $A$ to $B C$ at $T$. Given that $O I \| B C$, show that $P Q=P T$.