Circle Triangle Picking

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TriangleInscribing

Circle triangle picking considers the areas of triangles determined by three points chosen independently and uniformly at random on the circumference of a unit circle.

By rotational and reflection symmetry, the first point can be assigned coordinates and the central angles from it to the second and third points, and , can be taken as independent random variables with uniform distributions on and , respectively. The signed area is

(1)

and the ordinary area is , so

			(2)
			(3)

Therefore,

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

But

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

Write the mean area as

(12)

then

(13)

and

(14)

Evaluating the antiderivative above gives

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

so

(20)

Also,

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

so

(25)

Combining (20) and (25) gives the mean triangle area as

(26)

(OEIS A093582).

The first few raw 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 resulting triangle contains the origin is 1/4.

See also

Circle Line Picking, Disk Triangle Picking, Line Line Picking, Sphere Point Picking

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References

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

Referenced on Wolfram|Alpha

Circle Triangle Picking

Cite this as:

Weisstein, Eric W. "Circle Triangle Picking." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CircleTrianglePicking.html

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