Comment by xeonmc

I think it would be clearer with an explicit function of a rectangle's area with respect to its diagonals:

                   A = x y
                   x = r cos(θ)
                   y = r sin(θ)
             => A(r) = r²cos(θ)sin(θ)
                A(c) = A(a) + A(b)
    = c²cos(θ)sin(θ) = a²cos(θ)sin(θ) + b²cos(θ)sin(θ)
               => c² = a² + b²