the geometric mean

The geometric mean of two positive numbers is a kind of 'multiplicative middle' between them. Where the ordinary (arithmetic) average of 4 and 9 is found by adding and halving — getting 6.5 — the geometric mean is found by multiplying and taking the square root, getting the square root of 36, which is also 6. It answers the question: what single number, when used in proportion, sits between these two?

Formally, the geometric mean of a and b is the value x such that a : x = x : b — the repeated middle term of a proportion. By the means-extremes property this gives x^2 = a·b, so x = sqrt(a·b) (taking the positive root). For 4 and 9, x = sqrt(36) = 6. Geometrically, x is the side of a square whose area equals that of an a-by-b rectangle: the geometric mean turns a rectangle into an equal-area square. This is why the mean proportional and the geometric mean are the same idea under two names.

In this field the geometric mean is the heart of the right-triangle altitude relations: when you drop the altitude from the right angle to the hypotenuse, that altitude is the geometric mean of the two pieces it creates, and each leg is the geometric mean of the whole hypotenuse and the piece next to it. The geometric mean is always less than or equal to the arithmetic mean (equal only when a = b), a fact that quietly underlies many inequalities — but in geometry its job is to be the length that lives 'in proportion between' two others.