Here is an old geometry/measurement chestnut for you.

[Whateverist]

Well I misspoke in my first thoughts on the more general problem.

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What I should have said needed to be shown is that the sum of the semicircles was equal in area to the semicircle containing the inscribed triangle.

But that semicircle is just the reflection across the diameter of the semicircle on the inscribed circle's diameter, so I guess my work is done, since then the area of interest can be shown to be equal to the area of the triangle.

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This more general problem would seem considerably more complex without having considered the easier form first.