Is there a point at rational distances from the vertices of a unit square? This unsolved question is known as the rational distance problem [1, 2].

This Demonstration gives 2877 canonical triples, which are points at rational distances from the vertices , , and . These triples were collected by analyzing primitive Heronian triangles [3] (triangles with rational sides and areas). If is a triple, so are and the inverse , so each triple gives three others.

A triple has rational coordinates. Consider the squares of the distances: , , and All of these need to be rational, so the differences and are also rational.

The ellipse has 193 of the triples (or 386 counting reflections in ), with inverse points on the cubic .

Point 1269, , is also a rational distance from .

References

[1] R. K. Guy, "Rational Distances from the Corner of a Square," in Unsolved Problems in Number Theory, 2nd ed., New York: Springer-Verlag, 1994 pp. 181–185.