Ratio identity given A:B = C:D

Let $\frac{A}{B} = \frac{C}{D} = k \implies A = Bk, C = Dk$.

$\frac{A^2 + B^2}{C^2 + D^2} = \frac{B^2 k^2 + B^2}{D^2 k^2 + D^2} = \frac{B^2 (k^2 + 1)}{D^2 (k^2 + 1)} = \frac{B^2}{D^2}$

Also check $\frac{A \times B}{C \times D} = \frac{(Bk) \times B}{(Dk) \times D} = \frac{B^2 k}{D^2 k} = \frac{B^2}{D^2}$.

Thus, $\frac{A^2 + B^2}{C^2 + D^2} = \frac{A \times B}{C \times D}$.