Fisher's Theorem

Let $A$ be a sum of squares of $n$ independent normal standardized variates $x_i$, and suppose $A=B+C$ where $B$ is a quadratic form in the $x_i$, distributed as Chi-Squared with $h$ Degrees of Freedom. Then $C$ is distributed as $\chi^2$ with $n-h$ Degrees of Freedom and is independent of $B$. The converse of this theorem is known as Cochran's Theorem.

See also Chi-Squared Distribution, Cochran's Theorem