Another mean, the geometric mean is defined for non-negative values only. For x1,...,xn, it is the n'th root of (x1*...*xn).

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!