4
$\begingroup$

Given a circle with radius R = 1, I'm trying to find either the probability distribution function or the density function for the space of triangle, which is randomly selected on this circle. The same task is for perimeter function of this triangle.

enter image description here

The only thing I've understood is the following. If we fix some point R on the circle, then angles ROA, ROB, ROC (counterclockwise) are uniformly distributed on [0; 2 * Pi]. I've tried expressing the space and the perimeter through those angles, but still had no success.

I would appreciate any help, really. I've tried to solve this problem for three weeks, and it seems to me that soon those triangles and circles will begin to come into my night dreams. Thanks.

  • 0
    I get the impression there won't be a nice answer to this. There are various ways of expressing the area and the perimeter in terms of the angles, and you can get e.g. the average area, $3/(2\pi)$, by integrating over the angles, but as far as I'm aware, in order to get the distribution, you'd have to solve for one of the angles, and I invariably get ugly fourth-order equations when I do that.2011-12-17

6 Answers 6