Solution
The Largest Angle
Show the problem again
Three points are chosen independently and uniformly at random on a circle and joined to form a triangle. What is the expected size of the triangle's largest angle, in degrees?
Worked solution
The answer is 110°. By the inscribed angle theorem, each angle of the triangle is half the central angle of the arc opposite it. The three arcs sum to 360°, so the largest angle is half the largest arc. Cutting the circle at one of the three points turns the three arcs into the three pieces of a stick broken at two uniform points. The expected longest of n uniform pieces is (1/n)(1 + 1/2 + ... + 1/n), which for n = 3 is (1/3)(11/6) = 11/18. So the largest arc averages (11/18)·360° = 220°, and the largest angle averages 110°. So the "typical" random inscribed triangle is obtuse, consistent with the fact that three random points on a circle form an obtuse triangle three quarters of the time.
Source: Classic geometric-probability exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.