Geometric Mean of Reciprocals is Reciprocal of Geometric Mean

Theorem
Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers.

Let $G_n$ denote the geometric mean of $x_1, x_2, \ldots, x_n$.

Let ${G_n}'$ denote the geometric mean of their reciprocals $\dfrac 1 {x_1}, \dfrac 1 {x_2}, \ldots, \dfrac 1 {x_n}$.

Then:
 * ${G_n}' = \dfrac 1 {G_n}$