A random point of the circumdisk of a random triangle in a parallelogram

Abstract

For every parallelogram we evaluate in closed form the probability that a uniform random point of the circumdisk of a random triangle lies in the triangle, for three independent uniform vertices in the parallelogram. Its dilogarithm part has eleven functions with coefficients rational in two shape parameters, and no representation of this form uses fewer. For rectangles and rhombi it reduces to logarithms, arctangents, dilogarithms, the Clausen function and the inverse tangent integral. For the square it is 0.1625176680..., and up to similarity the square is the only maximizer among parallelograms; among rectangles the probability decreases strictly in the ratio ν of the longer to the shorter side, among rhombi it increases strictly in the smaller interior angle φ, and for (ν,φ) ∈ [1,16]×[π/60,π/2] it decreases strictly in ν and increases strictly in φ. The extremal and monotonicity results are computer-assisted. 2020 Mathematics Subject Classification: 60D05, 52A22, 33B30.