Fibonacci Identity

Since

$$\vert(a+ib)(c+id)\vert=\vert a+ib\vert\,\vert c+di\vert$$ (1)

$$\vert(ac-bd)+i(bc+ad)\vert = \sqrt{a^2+b^2} \sqrt{c^2+d^2},$$ (2)

it follows that

$$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(bc+ad)^2\equiv e^2+f^2.$$ (3)

This identity implies the 2-D Cauchy-Schwarz Sum Inequality.

See also Cauchy-Schwarz Sum Inequality, Euler Four-Square Identity

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, p. 9, 1996.