Let the incircle of triangle $ABC$ touch sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$, respectively. Let $Γ$,$Γ\_1$,$Γ\_2$ and $Γ\_3$ denote the
circumcircles of triangle $ABC$, $AEF$, $BDF$ and $CDE$ respectively. Let $Γ$ and $Γ\_1$ intersect at $A$ and $P$, $Γ$ and $Γ\_2$ intersect
at $B$ and $Q$, and $Γ$ and $Γ\_3$ intersect at $C$ and $R$.

Prove that $PEDQ$ is cyclic.

[![enter image description here](https://i.stack.imgur.com/HdtDO.png)](https://i.stack.imgur.com/HdtDO.png)