Sunday, October 4, 2009


What is the probability that three points, chosen at random from a circle, lie on a common semicircle?

1 comment:

  1. The answer is 3/4.

    Observe that if A, B, and C are three points on a circle, then the points lie on a common semicircle if and only if the center of the circle does not lie inside the triangle ABC.

    Let A, B, and C be three randomly selected points on the circle. Let a, b, and c be the points on the circle that are directly opposite to A, B, and C, respectively.

    Using these six points we form eight triangles: ABC, ABc, AbC, Abc, aBC, aBc, abC, and abc. Observe that exactly two of these triangles contain the center of the circle. But each triangle has the same likelihood of containing the center.

    Therefore, the probability that A, B, and C lie on a common semicircle is 6/8, or 3/4.