Let $ABC$ be a triangle, and let $X$, $Y$, and $Z$ be the excenters opposite $A$, $B$, and $C$. The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $D$, $E$, $F$, respectively. Finally, let $I$ and $O$ denote the incenter and circumcenter of triangle $ABC$, respectively.
Prove that lines $DX$, $EY$, $FZ$, $IO$ are concurrent.
My progress: I proved that there is a homothety that takes triangle $DEF$ to $XYZ$, proving that $DX$, $EY$, and $FZ$ concur. However I don't know how to prove the point of concurrency lies on $IO$. I do know that $IO$ is the Euler line of $XYZ$, but I haven't made more progress.