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.

Let a=vertical triangle side, b=horizonta triangle side, and c=hypotenuse; semi-a=red semi-circle, etc.

Area(crescents) = Area(total) - A(semi-c)

Area(crescents)=[Area(semi-a)+Area(semi-b)+Area(triangle)]-A(semi-c)

Area(crescents)=¨ö(¨ù¥ða©÷) + ¨ö(¨ù¥ðb©÷) + Area(triangle) - ¨ö(¨ù¥ðc©÷)

Area(crescents) = Area(triangle) + ¨ö(¨ù¥ð(a©÷+b©÷-c©÷))

Since c©÷=a©÷+b©÷, the third term is zero. Therefore,

Area(crescents) = Area(triangle)