Question

Suppose we draw semicircles on the sides of a triangle and compute their areas. If the sum of the areas of two of the semicircles equals the area of the third, is the triangle a right triangle?

Yes.
(pi/2)(a^2 + b^2) = (pi/2) c^2
if and only if
a^2 + b^2 = c^2,
and that requires a right angle triangle.