Metric Matrix in Barycentric Coordinates

TL;DR

The paper introduces a metric matrix $K$ to express inner products and angles in barycentric coordinates, turning planar relations into concise matrix identities. By constructing $K$ from $K_O$ and the Conway invariants, it derives a complete barycentric toolkit and proves key results such as Euler's relation $3G=2O+H$ and Feuerbach-type tangencies $|NI|= frac{1}{2}R-r$ and $|NI_A|= frac{1}{2}R+r_a$. The framework yields computational efficiency and deep geometric insight, with proposed extensions to lines, circles, conics, quadrilaterals, and potential applications in computer graphics and computational geometry.

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.

Theorems & Definitions (16)

• Theorem 1: Inner product formula
• Theorem 2: Oriented angle formula
• Theorem 3
• proof
• Theorem 4: Euler's theorem
• Theorem 5
• proof
• Theorem 6
• proof
• Theorem 7