Metric Matrix in Barycentric Coordinates

TL;DR

This paper introduces the metric matrix in barycentric coordinates to unify inner product formulation, revealing algebraic identities and deriving new proofs for classical triangle geometry theorems.

Contribution

It presents a novel metric matrix framework in barycentric coordinates, linking algebraic identities with geometric theorems and offering new proofs for classical results.

Findings

Unified formulation of inner product in barycentric coordinates

New algebraic identities related to barycentric coordinates

Novel proofs of classical triangle geometry theorems

Abstract

In this paper, the concept of the metric matrix is introduced to establish a concise and unified formulation for the inner product in barycentric coordinates. Building on this framework, we explore the intrinsic algebraic identities of barycentric coordinates and their direct correspondence with geometric theorems. Through this approach, we derive a series of novel results and provide new proofs for several classical theorems in triangle geometry.