Three points A, B, C are randomly distributed on a plane, with the following constrainsts:
- distance between A and B is equal to p
- distance between B and C is equal to q
What would be the average (expected value) for the distance r between A and C?
Of course the maximum is r = p + q. But what is the average? You can imagine A, B and C as being three cities. Without loss of generality, you can assume that A and B are fixed, and only C is random.
A nice answer was posted, involving some elliptic integral. Also an exact solution is available when p = q, and can be used in simulations to estimate pi (it's a stochastic geometry problem, like Buffon's needle).
Is there an exact solutions if the triangle is right? Does the average coincide with the case where the triangle right? (I believe so).
Anyway, read the answers (and contribute) at http://bit.ly/1fRVdGy
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