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

Geometric diagram of a right triangle ABC with its incircle (omega), tangent points D, E, F. It shows four semicircles on the hypotenuse AC (Red and Blue areas) and the two exterior Orange Lunes (lunulae) on the legs.

Problem statement

Let $ABC$ be a right triangle with the 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 $EH \perp AC$.

Given that 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 (lunulae) formed by constructing semicircles on the legs $AB$ and $BC$ exterior to the triangle, and a semicircle on the hypotenuse $AC$ passing through $B$.

*Note: Solving this problem may involve considering concepts such as the Circle and Pythagorean Theorem, Incircle and Right Triangle Segments, Equal Tangent Theorem, Right Triangle Area Identity, Lunes of Hippocrates Theorem.*

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