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# 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.

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 Web Resource. https://mathworld.wolfram.com/CircleTrianglePicking.html