$\quad \dfrac{a^2 + b^2}{c^2 + d^2}=\dfrac{ab}{cd}$
$\Leftrightarrow cd(a^2 + b^2)= ab(c^2 + d^2)$
$\Leftrightarrow a^2cd + b^2cd - abc^2 - abd^2 = 0$
$\Leftrightarrow (a^2cd - abc^2) + (b^2cd - abd^2)= 0$
$\Leftrightarrow ac(ad - bc)+ bd(bc - ad)= 0$
$\Leftrightarrow (ad-bc)(ac - bd)= 0$
$\Leftrightarrow \left[\begin{array}{l}ad - bc=0\\ac - bd=0\end{array}\right.$
$\Leftrightarrow \left[\begin{array}{l}ad = bc\\ac = bd\end{array}\right.$
$\Leftrightarrow \left[\begin{array}{l}\dfrac ab =\dfrac cd\\\dfrac ab =\dfrac dc\end{array}\right.$
