What about other “means”? It turns out these are also instances of the above mean, just in different codomains. To calculate f-mean (mean pulled by f), we map the values by function f into some other space, calculate the mean there, and pull back by f^{-1}.

For example, geometric mean is log-mean, and harmonic mean is reciprocal-mean. (Square mean is literally squaring-mean.) And these are useful in many applications, so they have special names.

Now I suppose it is clear what happened: you have calculated the angle-mean of all the chords, where angle is the function that maps the length of the chord to size of its central angle. You’ve mapped chord lengths to angle sizes, calculated the mean (very easily, by symmetry) in the angle-space, and pulled that mean (=0) to chord-space, yielding sqrt2.

So you can get almost any value in domain space (at least in the interval spanned by actual values) as f-mean for suitably chosen f. You have just chosen f so that the codomain mean can be easily calculated, not as some intrinsic property of chord lengths. But you don’t have to be too disappointed, since great mathematicians had almost the same problems in a slightly more difficult setting. See http://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

I think I was viewing the problem continuously, not discretely, but I’m still not wrapping my brain around why it would be a median but not a mean — I’m seeing symmetry, so the two should be the same. (But I think it’s quite possible I’m making some other assumptions, as jedwards points out. I was certainly assuming Euclidean, but being in a plane follows from that and the fact that there are three points. [Wait, is that right? Maybe the mean would change if C varies over a sphere rather than a circle. OK, I think I'm seeing how the mean and median might be different.]

@Mitchell: Can you explain what you see as the difference between a discrete and a continuous perspective on this problem?

Interesting question.

A simpler way of asking the same question would be what is the average length of a chord in a circle with a radius of 1.

Here’s my reasoning. If the angle between the segments is , then the distance between A and C is . Then the average value of this function on the interval is .