The Rational Distance Problem for Isosceles Triangles with one rational side

TL;DR

This paper fully characterizes isosceles triangles with one rational side for which points at rational distances to all vertices exist, advancing understanding of the Rational Distance Problem in specific geometric configurations.

Contribution

It provides a complete description of isosceles triangles with one rational side where the rational distance problem has solutions.

Findings

Solutions are dense in the plane for certain triangles.

Complete characterization of isosceles triangles with rational solutions.

Clarifies conditions under which the problem (P) is solvable.

Abstract

For a triangle $\Delta$, let (P) denote the problem of the existence of points in the plane of $\Delta$, that are at rational distance to the 3 vertices of $\Delta$. Answer to (P) is known to be positive in the following situation: $\Delta$ has one rational side and the square of all sides are rational. Further, the set of solution-points is dense in the plane of $\Delta$. See [3] The reader can convince himself that the rationality of one side is a reasonable minimum condition to set out, otherwise problem (P) would stay somewhat hazy and scattered. Now, even with the assumption of one rational side, problem (P) stays hard. In this note, we restrict our attention to isosceles triangles, and provide a \textit{complete description} of such triangles for which (P) has a positive answer.