Help me with this really hard math problem

Each side of an issocles right triangle is the diameter of a semicircle. What is the area of the triangle if the sum of the areas of the 3 semicicrcles is 200pi?
(please show work and explain)

Let
a and b be the 2 equal sides of the triangle
c be the hypotenuse
A be the area of the semicircle on a = 0.5*(pi)*(0.5a)^2
B be the area of the semicircle on b (A=B because a=b)
C be the area of the semicircle on c

We have that

a=b
a^2+b^2=c^2 <=> 2a^2=c^2
2*0.5*(pi)*(0.5a)^2 + 0.5*(pi)*(0.5c)^2=200 (pi)

We want to know the area of the triangle, given by 0.5*a^2

therefore
2*0.5*(pi)*(0.5a)^2 + 0.5*(pi)*(0.5c)^2=200 (pi) <=> 0.25 a^2 + 0.125*c^2=200 <=>
<=> 0.25 a^2 + 0.125 *2*a^2=200 <=> 0.5*a^2=200

Therefore, in the given conditions, the area of the triangle is 200.

The sides of the triangle are a, a, and a√2. So the radius of the three semicircles are a/2, a/2, and a/√2.

Sum of the areas pf the three semicircles = πa^2/8 + πa^2/8 + πa^2/4 = πa^2/2 = 200π
==> a^2/2 = 200 ==> a^2 = 400 ==> a = 20

Area of the triangle = a*a/2 = a^2/2 = 200