Definition:Geometric Mean

Definition
Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive.

The geometric mean of $x_1, x_2, \ldots, x_n$ is defined as:


 * $\displaystyle G_n := \left({\prod_{k \mathop = 1}^n x_k}\right)^{1/n}$

That is, to find out the geometric mean of a set of $n$ numbers, multiply them together and take the $n$th root.

Mean Proportional
In the language of Euclid, the geometric mean of two magnitudes is called the mean proportional.

Thus the mean proportional of $a$ and $b$ is defined as that magnitude $c$ such that:
 * $a : c = c : b$

where $a : c$ denotes the ratio between $a$ and $c$.

From the definition of ratio it is seen that $\dfrac a c = \dfrac c b$ from which it follows that $c = \sqrt {a b}$ demonstrating that the definitions are logically equivalent.

Note that this definition is never made specifically in Euclid's, but introduced without definition in the porism to Perpendicular in Right-Angled Triangle makes two Similar Triangle.

It is mentioned again, in the same context, in Construction of Mean Proportional.

Also see

 * Mean
 * Arithmetic Mean Never Less than Geometric Mean