New Proofs of Triangle Inequalities

TL;DR

This paper introduces three novel proofs of the triangle inequality in Euclidean Geometry, utilizing circles and ellipses instead of traditional triangle properties, aiming for simplicity.

Contribution

It provides new proofs of the triangle inequality based on circle and ellipse properties, diverging from the conventional triangle-based proofs.

Findings

Three new proofs of the triangle inequality are presented.

The proofs avoid using traditional triangle properties.

The methods involve circle and ellipse properties.

Abstract

We give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the triangle inequality as simple as possible without using properties of triangles.