Ceva's triangle inequalities

TL;DR

This paper characterizes triples of cevians that form a triangle regardless of the original triangle, by solving a specific family of inequalities called Ceva's triangle inequalities, and provides a parametrization of all solutions.

Contribution

It introduces a complete parametrization of solutions to Ceva's triangle inequalities, advancing understanding of cevian configurations independent of the original triangle.

Findings

Solution set characterized by a three-parameter family

Parametrization of cevian triples forming a triangle

Independent cevian triangle conditions established

Abstract

We characterize triples of cevians which form a triangle independent of the triangle where they are constructed. This problem is equivalent to solving a three-parameter family of inequalities which we call Ceva's triangle inequalities. Our main result provides the parametrization of the solution set.