KöMaL 2025 B5501

The incircle of triangle $ABC$ is tangent to sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $G$ be the antipode of point $F$ on the incircle. Lines $GD$ and $GE$ intersect line $AB$ at points $D'$ and $E'$, respectively. Prove that circles $AED'$ and $BDE'$ intersect each other on the incircle.