Assume circle with arc AXC, Area = (1/2b)(1/2b)*pi

Assume circle with arc CYB, Area = (1/2a)(1/2a)*pi

Area Circle inscribing ABC less area of triangle ABC, Area = (1/2c)(1/2c)*pi-1/2ab

Shaded area = AXC + CYB -(Area ACB - Area triangle)

A = (b*b/4)pi +(a*a/4)pi - (c*c/4)pi+ab/2

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

Substitute c*c for (b*b + a*a) (Pythagorean theorum)

= pi/4(c*c-c*c) +ab/2 = ab/2