"What is the area? You should be able to solve this"

[Pekis]

Hello,
Follow me in my complicated way:
A=16
B=20
C=32
X=?

If d is Big Square side:
=> d*d=A+B+C+X

Surface X=Rectangle+UpTriangle+DownTriangle
Rectangle=(d/2-b)*d/2 (b is base of DownTriangle X)
UpTriangle=a*(d/2-b)/2 (a is base of UpTriangle X)
DownTriangle=b*(d/2+a)/2
=> X=d/4*(d+a-b)

Surface A=Rectangle+UpTriangle+DownTriangle
Rectangle=(d/2-a)*(d/2-b)
UpTriangle=UpTriangle X
DownTriangle=b*(d/2-a)/2
=> A=d/4*(d-a-b)

Surface B=Rectangle+UpTriangle+DownTriangle
Rectangle=(d/2-a)*d/2
UpTriangle=DownTriangle A
DownTriangle=a*(d/2+b)/2
=> B=d/4*(d-a+b)

Surface C=Rectangle-DownTriangle B-DownTriangle X
Rectangle=(d/2+a)*(d/2+b)
=> C=d/4*(d+a+b)

Sanity check of A+B+C+X:
d/4*(d-a-b)+d/4*(d-a+b)+d/4*(d+a+b)+d/4*(d+a-b) simplifies as d*d, without relation to a and b, as expected

Now,
C=2*A => d=3*(a+b)
4*B=5*A => d=a+9*b
5*C=8*B => d=13*a/3-b
Solve system of 3 equations with 3 unknowns
=> a is free, b=a/3, d=4*a

Pushing this back into earlier formula A=d/4*(d-a-b)
=> a=sqrt(6)
=> b=sqrt(6)/3, d=4*sqrt(6)

As X=d*d-A-B-C
=> X=(4*sqrt(6))^2-16-20-32-=28

Delightfully complicated