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| Math Central | Quandaries & Queries |
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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. |
Greg,
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?
Penny
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