The intouch triangle and excentral triangle

Question

Let $I_a$, $I_b$, and $I_c$ be the excenters of triangle $ABC$, and let $D$, $E$, $F$ be the intersection points of the incircle to segments $BC$, $AC$, and $AB$. Prove that the center of homothety taking $DEF$ to $I_aI_bI_c$ lies on line $OI$, where $O$ is the circumcenter of $ABC$ and $I$ is the incenter of $ABC$.