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- vanship November 30, 2011Denote the first two points A and B that make the longest edge of the triangle, the third one is C, and O is the center of the circle. Also let the circumstance of the circle to be 1, the shorter curve AB is x, where 0<x<=1/2.
First, the probability that edge AB are the diameter, where x=1/2 is 0, so the prob for right angle triangle is 0.
Second, to make an obtuse triangle, and given the assumption that AB is the longest edge, C can only lie on the shorter curve AB on the circle. Otherwise AB will not be the longest edge or ABC is not an obtuse triangle. This gives the possibilities of
2*int_{x=0}^{1/2} x = 1/4, where the factor of 2 is due to symmetry.
Third, to make an acute triangle, C can be only chosen under the condition that angle AOC < angle AOB, and angle BOC < angle AOB. Otherwise, AB will not be the longest edge. Translated to the curve lengths, given shorter curve AB = x, we have 1-2x < curve AC < x. So x should be greater than 1/3 and the length of possible C, given B, is
x - (1-2x) = 3x-1.
The total possibilities are
2*int_{x=1/3}^{1/2} 3x-1 = 1/12, where the factor of 2 is due to symmetry.
So the possibilities of obtuse and acute triangles are 1/4:1/12 = 3:1. So the probabilities for them are 3/4 and 1/4 respectively.