Several special cases of a square problem

TL;DR

This paper investigates special cases of the square problem, showing that certain points within a square cannot have four rational distances to the vertices, supporting the conjecture that such a point may not exist.

Contribution

The paper proves that for specific configurations within a square, the distances to vertices cannot all be rational, extending the understanding of the rational distance problem.

Findings

Distances on diagonals, midlines, or edges are not all rational.

Special side-length conditions (prime multiples) prevent rational distances.

Results can be extended from the square to the entire plane.

Abstract

Here is a square problem: in a unit square, is there a point with four rational distances to the vertices? A probability argument suggests a negative answer. This paper proves several special cases of the square problem: if the point sits on the diagonals, the midlines or the edges of the square, or the side-length of the square is n times the distance from the point to one side (both $n$ and $(n^2+4)$ are prime numbers), the distances from this point to the four vertices can not be all rational. However, this paper does not prove a more general situation. The proof here can be extended to the whole plane, instead of being limited to the interior of the square. Key Words: discrete Geometry; rational distance; a square problem.