Does the triangle contain the center?

Question

Three points are placed independently and uniformly at random on the circumference of a circle, and joined to form a triangle. What is the probability the triangle contains the center of the circle?

Accepted Answer

The key geometric fact: the triangle formed by three points on a circle contains the center if and only if the three points do not all lie in a semicircle. If they all fit in one half, the whole triangle sits on that side and the center is outside it; if they straddle both halves, the center is enclosed. So the two events are complements: P(	ext{center inside}) = 1 - P(	ext{all three in a semicircle}). From the semicircle result, three uniform points all lie in some semicircle with probability 	frac{3}{2^{2}} = 	frac34. Therefore P(	ext{center inside}) = 1 - \frac34 = \frac14. To see the…

Does the triangle contain the center?

Your answer

More Probability questions