KöMaL 2025 B5431

Let $k$ denote the incircle of triangle $ABC$, and let $I$ denote its center. Line $AI$ intersects circle $k$ in points $D$ and $E$ such that $D$ is closer to $A$ than $E$. Let $k$ be tangent to $AC$ at $F$. Let the tangent of $K$ at point $E$ intersect $AC$ at $G$. Prove that $GI$ is parallel to $FD$.