A Metric Framework for Triangle Inequalities via Barycentric Coordinates

TL;DR

This paper introduces a unified metric framework using barycentric coordinates to derive and refine triangle inequalities, providing geometric insights and extending to higher dimensions and non-Euclidean spaces.

Contribution

It develops a novel metric-based approach to triangle inequalities with barycentric coordinates, unifying and generalizing classical results.

Findings

Derived new generalized inequalities involving the incenter and Euler line

Refined existing classical triangle inequalities

Extended the framework to higher-dimensional and non-Euclidean geometries

Abstract

This paper presents a unified metric-based framework for triangle geometric inequalities using barycentric coordinates. By interpreting classical inequalities as squared distances between points(a process termed metricization)we derive and refine numerous well-known inequalities. Furthermore, the squared-distance function ELD_I, measures distances from the incenter to points on the Euler line, also yielding generalized inequalities. The method offers geometric clarity and extends naturally to higher-dimensional and non-Euclidean settings.