Math CentralQuandaries & Queries


Question from Greg:

Points A1, A2, . . ., AN are equally spaced around the circumference of a circle and N >=3. Three of these points are selected at random and a triangle is formed using these points as its vertices. If N = 2k for some positive integer k >= 2, determine the probability that the triangle is acute.


Here's some hints on how to get going. First observe that if two of your points are diagonally opposite you do not have an acute triangle - what is it? So any choice of 3 points with a diagonally opposite pair have to be excluded. For the rest, look at one of your points and the diagonally opposite point (which is not among your selection). Your other 2 points are either both on one side of this diagonal or one on each side - which corresponds to an acute triangle?



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