Heron isosceles with integral external radii

TL;DR

This paper derives formulas for external circle radii of triangles, focuses on Heron triangles with integer sides and area, and parametrically describes Heron isosceles triangles with integral external radii, providing examples and tables.

Contribution

It introduces a parametric description of Heron isosceles triangles with integral external radii, expanding understanding of their geometric properties.

Findings

Formulas for external radii in terms of side lengths

Parametric descriptions of Heron isosceles triangles

Identification of triangles with integral external radii

Abstract

Each triangle has three exterior or external circles tangential to the three straight lines containing the three sides of the triangle.Among the preliminaries in this paper, is deriving formulas for the radii of the three exterior circles in terms of the triangle's sidelengths. After that we focus on Heron triangles. Heron triangles are known in the literature as triangles with integer sidelengths and integral area.Pythagorean triangles are examples of Heron triangles. In the first part of the paper, we parametrically describe all Heron isosceles triangles. In the second part, we parametrically(in terms of three independent parameters) describe the subfamily which consists of all Heron isosceles triangles which also have integral external radii. We provide numerical examples and tables.