KöMaL 2023 C1782

From vertex $D$ of a square $ABCD$, a tangent is drawn to the semicircle of diameter $AB$ lying inside the square. The point of tangency (different from point $A$) is $E$, and the midpoint of line segment $AB$ is $F$. The tangent intersects line $BC$ at point $G$ and line $AB$ at point $H$. Line $EF$ intersects line $DA$ at point $K$. Prove that the radii of the inscribed circles of triangles $FHE$, $DGC$ and $DKE$, in this order, form three consecutive terms of an increasing arithmetric sequence.