Rational Distance Problem

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.


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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 .
[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.
[2] Wikipedia. "Unit Square." (May 26, 2016) en.wikipedia.org/wiki/Unit_square.
[3] S. Kurz. "Heronian Figures." (May 26, 2016) www.wm-archive.uni-bayreuth.de/index.php?id=554&L=3.
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