To show: The white area inside the figure is equal to the dark gray area.
We first observe that that the entire figure is one quarter of a circle of radius r and that the two semicircles taken together form a circle of radius r/2.
We're now ready to write a couple of equations:
area of entire figure = (white area) + (light gray areas) + (dark gray area) = (1/4)πr2
area circle of radius r/2 = (light gray areas) + 2(dark gray area) = π(r/2)2 = (1/4)πr2
Subtracting the second equation from the first, we have
(white area) − (dark gray area) = 0
(white area) = (dark gray area)
Source: The Tokyo Puzzles by Kobon Fujimura