Bulging Triangles: Generalization of Reuleaux Triangles

TL;DR

This paper introduces a new class of geometric shapes called bulging triangles, generalizing Reuleaux triangles, and explores their properties, including whether fundamental triangle inequalities and the Pythagorean theorem apply.

Contribution

It presents a novel generalization of Reuleaux triangles with vertices fixed, and investigates their geometric properties and fundamental theorems using calculus.

Findings

Bulging triangles can be constructed with fixed vertices.

Basic triangle inequalities may or may not hold for these shapes.

The Pythagorean theorem's applicability is examined for bulging triangles.

Abstract

We introduce a bulging triangle like the generalization of the Reuleaux triangle. We may be able to propose various ways to bulge a triangle, but this paper presents the way so that its vertices are the same as them of the original triangle. We find some properties and theorems of our bulging triangles. In particular, we investigate, via calculus, whether basic facts such as triangle inequalities and Pythagorean theorem hold for bulging triangles.