Lunes

The word “lune” takes its origin from the Latin word luna, meaning “moon.” In geometry, lunes are plane regions bounded by arcs of different circles (see shaded crescents in the figure below). Prove that the sum of the areas of two lunes constructed on two sides of a triangle which is inscribed in a semi-circle equals the area of that triangle.

That is, in the above figure, if ABC, AEB and BFC are semicircles, prove that the area of Lune1 + the area of Lune 2 = the area of the triangle ABC.

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