The Rational Distance Problem for Equilateral Triangles

TL;DR

This paper completely solves the Rational Distance Problem for equilateral triangles, determining when a point exists in the plane at rational distances from all vertices.

Contribution

It provides a complete characterization of the Rational Distance Problem specifically for equilateral triangles, a case not fully addressed before.

Findings

Characterization of points with rational distances in equilateral triangles

Conditions under which such points exist

Complete solution to the problem for all equilateral triangles

Abstract

Let (P) denote the problem of existence of a point in the plane of a given triangle T, that is at rational distance from all the vertices of T. In this article, we provide a complete solution to (P) for all equilateral triangles.