Given a right triangle, draw the three following semicircles:

- The semicircle with diameter formed by one of the legs and extending away from the triangle.

- The semicircle with diameter formed by the other leg and extending away from the triangle.

- The semicircle with diameter formed by the hypotenuse and extending towards the triangle.

Prove that the area of the two crescents (shown in **RED** and **BLUE**) formed by the three semicircles equals the area of the triangle.

Be S de desired area value. Be 2a the smaller side, 2c the hypotenusa and 2b the other side of the triangle.

So the area of the triangle is (4ab)/2.

The area of the first semicircle is (pi*(a^2))/2.

The area of the second semicircle is (pi*(b^2))/2.

The area of the third semicircle is (pi*(c^2))/2.

The desired area is

(pi*(a^2))/2 + (pi*(b^2))/2 - ((pi*(c^2))/2 - (4ab)/2).

But 4(c^2) = 4(a^2) + 4(b^2) (right triangle), so we have

(c^2) = (a^2) + (b^2)

The desired area is (4ab)/2, the same as the triangle area.