CMO 2007 Problem 5

Let the incircle of triangle $ABC$ touch sides $BC, CA$ and $AB$ at $D, E$ and $F,$ respectively. Let $\omega,\,\omega_{1},\,\omega_{2}$ and $\omega_{3}$ denote the circumcircles of triangle $ABC, AEF, BDF$ and $CDE$ respectively.

Let $\omega$ and $\omega_{1}$ intersect at $A$ and $P,\,\omega$ and $\omega_{2}$ intersect at $B$ and $Q,\,\omega$ and $\omega_{3}$ intersect at $C$ and $R.$

$a.$ Prove that $\omega_{1},\,\omega_{2}$ and $\omega_{3}$ intersect in a common point.

$b.$ Show that $PD, QE$ and $RF$ are concurrent.