KöMaL 2025 B5480

In triangle $ABC$ angle $\angle ACB$ is $90^{\circ}$. Let $F$ denote the midpoint of leg $AC$, and let $G$ denote the midpoint of hypotenuse $BC$. Prove that the circle with diameter $FG$ is tangent to the incircle of triangle $ABC$ if and only if $AC=2BC$.