Eight Selections in Six Sectors

Eight Selections in Six Sectors
What is this about?
A Mathematical Droodle

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The applet attempts to suggest an argument in favor of the thesis that there are 3 times as many obtuse triangles as there are acute ones.

Three points A, B, C are distributed uniformly on a circle. What is the probability of their forming an obtuse triangle?

Eight Selections in Six Sectors

N. Vasiliev credits V. V. Fok and Yu. V. Chekanov with the following argument.

We shall select a triangle in two steps. First we get three points A, B, C and their antipodes A', B', C' across the center of the circle. Selecting a triangle consists in selecting one of the points in each of three pairs {A, A'}, {B, B'}, {C, C'}. There are 8 possible combinations.

With each of the six points we associate the semicircle around its antipode. Three points form an obtuse triangle if and only if they lie in a semicircle and this is only if their associated semicircles have a nonempty intersection. The diameters of the associated semicircles form 6 sectors. Each sector serves as an intersection of the associated semicircles for exactly one selection of the vertices, meaning that out of 8 possible selections there are always 6 obtuse (and 2 acute) triangles. The probability that one is obtuse is therefore,

6/8 = 3/4.

References

N. Vasiliev, Geometric Probability, Kvant (Russian), #1, 1991, 48-53

Geometric Probability

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