What is the probability that three random points on a unit circle would form a triangle that includes the center of the unit circle?

Select three points at random on the circumference of a unit circle and find the distribution of areas of the resulting triangles determined by these three points.

The first point can be assigned coordinates

without loss of generality. Call the central angles from the first point to the second and third and . The range of can be restricted to because of symmetry, but can range from . Then

so

Therefore,

But

Write (10) as

then

and

From (12),

so

Also,

so

Combining (◇) and (◇) gives the mean triangle area as

(OEIS A093582).

The first few moments are

(OEIS A093583 and A093584 and OEIS A093585 and A093586).

The variance is therefore given by

The probability that the interior of the triangle determined by the three points picked at random on the circumference of a circle contains the origin is 1/4.