Another mean, the *geometric mean* is defined for non-negative values only. For x_{1},...,x_{n}, it is the n'th root of (x_{1}*...*x_{n}).

The AM-GM inequality relates the geometric mean to the more famous arithmetic mean (and also to the lesser known harmonic mean).

## So what's so geometric about this mean?

Place two line segments, of lengths a and b, adjacent on the same line. Construct a circle with the line segment a+b as its diameter. Obviously, its radius is the arithmetic mean (a+b)/2. BUT...

Raise a perpendicular from between a and b to the circle. The length of this perpendicular is precisely sqrt(ab) -- the *geometric mean* of a and b!