equilateral triangle line picking | isosceles right triangle line picking

Consider the average length of a line segment determined by two points picked at random in the interior of an arbitrary triangle Δ(a, b, c) with side lengths a, b, and c. 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 ℓ^__(Δ(a, b, c)) | = | (4s s_a s_b s_c)/15[1/a^3 ln(s/s_a) + 1/b^3 ln(s/s_b) + 1/c^3 ln(s/s_c)] + (a + b + c)/15 + ((b + c)(b - c)^2)/(30a^2) + ((c + a)(c - a)^2)/(30b^2) + ((a + b)(a - b)^2)/(30c^2)

equilateral triangle | isosceles right triangle | square line picking | triangle point picking | triangle triangle picking