Archimedes' twin circles

The twin circles of Archimedes are two circles that are in a Arbelos inscribed (also known as Sickle of Archimedes called ).

The construction can be found in the Book of Lemmas, whose attribution to Archimedes is questionable.

Using the Pythagorean theorem can be shown that the two circles have the same radius, and thus the same area. Denoting the radii of the two smaller semicircles with and so applies to the radius of the twin circles: