Brahmagupta-Fibonacci Identity

Theorem
Let $a, b, c, d$ be numbers.

Then:

This is an example of a more general identity:

Proof
Setting $b = -b$ in the above gives the identity:


 * $\left({a^2 + n b^2}\right) \left({c^2 + n d^2}\right) = \left({a c - n b d}\right)^2 + n \left({a d + b c}\right)^2$

The identities:

follow from the above by setting $n = 1$.

Note
This identity is also known as Fibonacci's Identity, and is a special case for $n = 2$ of Lagrange's Identity.

Both of these described this identity in their writings:


 * 628: Brahmagupta: Brahmasphutasiddhanta (The Opening of the Universe)
 * 1225: Fibonacci: Liber quadratorum (The Book of Squares)