Catalan's Identity

Theorem
Let $F_k$ be the $k$th Fibonacci number.

Then:
 * ${F_n}^2 - F_{n - r} F_{n + r} = \left({-1}\right)^{n - r} {F_r}^2$

Also see

 * Cassini's Identity, the special case of this where $r = 1$