Some diophantine problems concerning a pair of rational triangles with a common circumradius

TL;DR

This paper investigates three new problems involving pairs of rational triangles sharing a common circumradius and either perimeter, inradius, or area, providing parametric solutions for each case.

Contribution

It introduces and solves three previously unstudied problems about pairs of rational triangles with shared circumradius and other properties, offering parametric solutions.

Findings

Parametric solutions for triangles with common circumradius and perimeter

Parametric solutions for triangles with common circumradius and inradius

Parametric solutions for triangles with common circumradius and area

Abstract

A triangle with rational sides and rational area is called a rational triangle. In this paper we consider three problems of finding pairs of rational triangles which have a common circumradius as well as either a common perimeter or a common inradius or a common area. While several similar problems concerning pairs of rational triangles have been considered in the existing literature, these three problems have not been studied till now. For each of these problems, we give a parametric solution and we also indicate how additional parametric solutions of these problems may be obtained.