Broken stick problem

In geometric probability, the broken stick problem asks for the probability that one can form a triangle from the three parts of a line segment that has been split randomly into three parts. Three line segments form a triangle if and only if they obey the triangle inequality: each segment's length is less than the sum of the other two lengths.¹ The problem was posed by Emanuel Czuber in 1884.² As with other problems in geometric probability, the answer depends on how one formalizes this random split using a probability distribution.¹

For two split points that are chosen uniformly and independently in the given line segment, the three lengths can be represented as the three barycentric coordinates of a point in an equilateral triangle whose height is the length of the original line segment, or equivalently as the three distances of the point from the sides of the triangle.¹ (This is an application of Viviani's theorem.) For this two-dimensional interpretation of the problem, found by Henri Poincaré,² the uniform distribution of the two split points corresponds to a uniform distribution of points within the equilateral triangle, and a pair of split points that forms a triangle corresponds to a point within the medial triangle of the given triangle. The area of the medial triangle is ${\displaystyle {\tfrac {1}{4}}}$ the area of the outer equilateral triangle, and so for this distribution of split points, the probability of forming a triangle is ${\displaystyle {\tfrac {1}{4}}}$.¹

An alternative two-dimensional interpretation uses the two split points as the Cartesian coordinates of a point in the unit square; the region of the unit square in which the two splits form three segments of a triangle is a crossed square with its vertices at the midpoints of the unit square, again having area ${\displaystyle {\tfrac {1}{4}}}$ of the total.³

An alternative probability distribution chooses the first split point uniformly at random, chooses one of the two resulting line segments randomly (with probability ${\displaystyle {\tfrac {1}{2}}}$ of choosing each segment), and then chooses a second split point uniformly at random within the chosen segment. For this model of the problem, the probability of forming a triangle is ${\displaystyle {\tfrac {1}{6}}}$.¹

A generalization of the problem asks whether a line segment broken into ${\displaystyle k}$ pieces can form a ${\displaystyle k}$-gon.⁴⁵