Consider the average length of a line segment determined by two points picked at random in the interior of an arbitrary triangle with side lengths , , and . This problem is not affine, but a simple formula in terms of the area or linear properties of the original triangle can be found using Borel's overlap technique to collapse the quadruple integral to a double integral and then convert to polar coordinates, leading to the beautiful general formula
(1)
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(A. G. Murray, pers. comm., Apr. 4, 2020), where is the semiperimeter and . The formulas for odd moments have a similar form to that of the mean but with higher powers of , , , and the triangle area .
This formula immediately gives the special cases obtained below originally using brute-force computer algebra.
If the original triangle is chosen to be an equilateral triangle with unit side lengths, then the average length of a line with endpoints chosen at random inside it is given by
(2)
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(3)
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The integrand can be split up into the four pieces
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As illustrated above, symmetry immediately gives and , so
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With some effort, the integrals and can be done analytically to give the final beautiful result
(9)
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(10)
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(OEIS A093064; E. W. Weisstein, Mar. 16, 2004).
If the original triangle is chosen to be an isosceles right triangle with unit legs, then the average length of a line with endpoints chosen at random inside it is given by
(11)
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(12)
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(13)
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(14)
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(OEIS A093063; M. Trott, pers. comm., Mar. 10, 2004), which is numerically surprisingly close to .
The mean length of a line segment picked at random in a 3, 4, 5 triangle is given by
(15)
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(16)
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(17)
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(E. W. Weisstein, Aug. 6-9, 2010; OEIS A180307).
The mean length of a line segment picked at random in a 30-60-90 triangle with unit hypotenuse was computed by E. W. Weisstein (Aug. 5, 2010) as a complicated analytic expression involving sums of logarithms. After simplification, the result can be written as
(18)
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(19)
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(E. Weisstein, M. Trott, A. Strzebonski, pers. comm., Aug. 25, 2010; OEIS A180308).
The expected distance from a random point in a general triangle to the vertex opposite the side of length is
(20)
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(A. G. Murray, pers. comm., Apr. 4, 2020), with analogous expressions for and . These give the beautiful identity
(21)
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(A. G. Murray, pers. comm., Apr. 4, 2020).