Prove equality using sum of squares and determinants Let $a$, $b$, $c$, $d$ be four positive real numbers satisfying that $a^2+b^2 = S = c^2+d^2$. I want to show that, under that hypotheses, it also holds:
$$ (ab+cd)S = (ac+bd)(ad+bc) $$
It seems to me that this equality can be deduced using suitable determinants of matrices. At a certain point one should use that $a^2+b^2$ and $c^2+d^2$ coincide.

You can simply calculate this algebraically, no need to consider matrices:
$$\begin{align*}(ac+bd)(ad+bc) &= a^2cd + abc^2 + abd^2 + b^2cd \\
&=  abc^2 + abd^2 +a^2cd + b^2cd \\ &= ab(c^2+d^2) + cd(a^2+b^2)\\
&= abS+cdS \\
&= (ab+cd)S\end{align*}$$
