A Geometric-Probabilistic problem about the lengths of the segments intersected in straights that randomly cut a triangle

If a line cuts randomly two sides of a triangle, the length of the segment determined by the points of intersection is also random. The object of this study, applied to a particular case, is to calculate the probability that the length of such segment is greater than a certain value.

💡 Research Summary

The paper tackles a classic geometric‑probability problem: given a triangle, a straight line is drawn at random so that it intersects two of the triangle’s sides, thereby creating a segment whose length is a random variable. The central question is to determine the probability that this segment exceeds a prescribed length L₀.
The authors begin by formalising what “random line” means in a planar setting. They parameterise a line by its orientation θ (0 ≤ θ < π) and by the position p of its intersection with a chosen reference side, measured as a fraction of that side’s length. Both parameters are assumed to be independent and uniformly distributed. This choice guarantees that every admissible line is equally likely, and it leads to a simple two‑dimensional probability space (