Intuition behind the Geometric Mean Our (awesome) statistics professor told us about the best intuition behind the definition of the Arithmetic Mean (he had heard of during his career). Here's what he said:

Imagine a 10-yard-long wooden plank. Think of each data value in the data set as a stone that weighs 1 pound with its position on the plank determined by the data value the stone represents. Now, place a fulcrum under the plank. The AM is the number corresponding to the fulcrum's position when the plank with the stones on it is completely still and parallel to the ground. The Weighted AM is the same situation with some of the stones having the weight distinct from 1 pound.

Of course, he then told us that this real-life physical example can easily be turned into a precise problem from physics and investigated formally using mathematics, arriving at the familiar nice properties the AM has.
He said he'd never seen a rationele for the definition of the Geometric Mean (or other means, for that matter) even remotely close to the one described above for the AM.
I thought I'd turn to the massive knowledgeable community of MSE for an answer.
Side question: besides the physical intuition described above, what other rationales are there for why the AM is defined the way it is?
 A: These examples are not both geometrical, but they are from real applications often encountered by statisticians, so perhaps they are intuitive.
Geometric mean. Suppose I make measurements $X_i$ that are strongly
right-skewed and experience has shown that it is easier to deal with
$Y_i = \log X_i.$ (One example is 'lognormal data' which you can read about on Wikipedia.) If I take the
arithmetic mean $\bar Y$ of the $Y_i,$ what does that correspond to in
terms of the original $X_i$? The answer is the log of their geometric mean:
$$ \bar Y = \frac{1}{n}\sum \log X_i = \frac{1}{n} \log \prod X_i =
 \log \left(\prod X_i\right)^{1/n}.$$
Harmonic mean. (As a bonus, to finish the usual triad of arithmetic, geometric, and harmonic means.) In the US, fuel efficiency of vehicles is usually
measured in miles per gallon (MPG). Suppose I drive downhill for a mile
at 40 MPG, drive round for a mile on flat roads at 30 MPG, and drive back
up the hill at 20 MPG. What is my average MPG? I have used 1/40 of a gallon, 1/30 of a gallon and then 1/20 of a gallon to go three miles. So gallons
per mile are about $$(.025 + .033 + .050)/3 = 0.036,$$
and the 'appropriate average' MPG is
$$\frac{3}{1/40 + 1/30 + 1/20} = 27.7,$$
which is the harmonic mean of the three separate MPGs. 
By contrast, in the metric system, fuel efficiency is usually measured
by distance per liter, and the arithmetic mean works fine.
A: Given a rectangular region of the plane measuring x units by y units, the geometric mean of x and y is the length of as side of the square with equal area to your rectangular region. 
