One circle is constructed on each side of a right triangle. The center of each circle is the midpoint of the side and the side forms a diameter of the circle. The area of the triangle is 24 square units. Find the total area of the regions of the two smaller circles that lie outside the largest circle.
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The area is $24$ square units. The triangle is inscribed into the circle constructed over the hypotenuse. It is contained in the semicircle to one side of the hypotenuse. The inner semicircles over the legs are entirely contained within the circle over the hypotenuse. By the Pythagorean theorem, the areas of the outer semicircles over the legs add up to the area of the semicircle over the hypotenuse that contains the triangle. The parts of the semicircle over the hypotenuse that extend beyond the triangle are precisely the parts of the outer semicircles over the legs that don't extend beyond the circle over the hypotenuse. It follows that the parts of the outer semicircles over the legs that do extend beyond the circle over the hypotenuse have the same area as the triangle itself.