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

TL;DR

This paper investigates the probability distribution of segment lengths formed when a random line intersects two sides of a triangle, focusing on calculating the likelihood that the segment exceeds a given length.

Contribution

It introduces a geometric-probabilistic framework for analyzing segment lengths in random line-triangle intersections, providing new methods for probability calculations.

Findings

Derived formulas for the probability that segment lengths exceed a threshold

Established relationships between line position and segment length distribution

Applied the model to specific triangle configurations

Abstract

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.