proving expression can be square

[gortwell]

Hi,

If it is known that \(\displaystyle (A^2+B^2)(C^2+D^2)\) is a perfect square
where A, B, C and D are integers, A and B are coprime, C and D are coprime and D > C >= B > A > 0

then \(\displaystyle (A^2+B^2)(C^2+D^2)+4(A^2-B^2)CD\) can be a perfect square...............(1)
and \(\displaystyle (A^2+B^2)(C^2+D^2)-4(A^2-B^2)CD\) can be a perfect square................(2)

but can it be proven that (1) and (2) can simultaneously be perfect squares or that they can't ?

thank you

examples

A = 1, B = 2, C = 2, D = 11
\(\displaystyle (A^2+B^2)(C^2+D^2)=25^2\)
\(\displaystyle (A^2+B^2)(C^2+D^2)+4(A^2-B^2)CD=19^2\)
\(\displaystyle (A^2+B^2)(C^2+D^2)-4(A^2-B^2)CD\ not\ a\ perfect\ square\)

A = 1, B = 2, C = 58, D = 209
\(\displaystyle (A^2+B^2)(C^2+D^2)=485^2\)
\(\displaystyle (A^2+B^2)(C^2+D^2)+4(A^2-B^2)CD\ not\ a\ perfect\ square\)
\(\displaystyle (A^2+B^2)(C^2+D^2)-4(A^2-B^2)CD=617^2\)