Rational Distance Problem
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.
(0,0)
(1,0)
(0,1)
(a,b)
(b,a)
(a,b)(+)
2
a
2
b
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.
2
x
2
y
2
(x-1)
2
y
2
x
2
(y-1)
2x-1
2y-1