Geometry Problem 1608: The Lunes of Hippocrates — Area via Incenter Semicircles

Problem statement

Let $ABC$ be a right triangle with right angle at vertex $B$. Let $\omega$ be the incircle of $\triangle ABC$, tangent to sides $AB$, $BC$, and $AC$ at points $D$, $E$, and $F$, respectively.

Let $G$ and $H$ be points on the hypotenuse $AC$ such that $DG \perp AC$ and $EH \perp AC$.

The sum of the areas of the semicircles constructed on $AG$ and $DG$ as diameters is $25\pi$, and the sum of the areas of the semicircles constructed on $EH$ and $CH$ as diameters is $4\pi$.

Determine the sum of the areas of the two lunes formed by constructing semicircles on the legs $AB$ and $BC$ exterior to the triangle, and a semicircle on the hypotenuse $AC$ passing through $B$.