TL;DR
This paper derives the probability distribution of the area of a triangle formed by randomly breaking a line segment twice, providing explicit formulas and median area calculations, and discusses related cyclic quadrilaterals.
Contribution
It determines the exact PDF and CDF of the triangle area from random breaks, including median area, and explores related geometric probability questions.
Findings
Median triangle area is approximately 0.031458 for segment length 1.
Probability that three pieces form a triangle is 1/4.
Explicit elliptic integral formulas for the distribution are derived.
Abstract
Breaking a line segment L in two places at random, the three pieces can be configured as a triangle T with probability 1/4. We determine both the PDF and CDF for area(T) in terms of elliptic integrals. In particular, if L has length 1, then the median area 0.031458... can be calculated to arbitrary precision. We also mention the analog involving cyclic quadrilaterals -- with corresponding probability 1/2 -- and ask some unanswered questions.
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Taxonomy
TopicsMathematics and Applications · Computational Geometry and Mesh Generation · Advanced Numerical Analysis Techniques