Circle Triangle Picking

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

(1)

so

			(2)
			(3)

Therefore,

			(4)
			(5)
			(6)
			(7)

But

			(8)
			(9)
			(10)
			(11)

Write (10) as

(12)

then

(13)

and

(14)

From (12),

			(15)
			(16)
			(17)
			(18)
			(19)

so

(20)

Also,

			(21)
			(22)
			(23)
			(24)

so

(25)

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

(26)

(OEIS A093582).

The first few moments are

			(27)
			(28)
			(29)
			(30)
			(31)
			(32)

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

The variance is therefore given by

(33)

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.

REFERENCES:

Sloane, N. J. A. Sequences A093582, A093583, A093584, A093585, and A093586 in "The On-Line Encyclopedia of Integer Sequences."