Rational Distance Problem

The rational distance problem asks to find a geometric configuration satisfying given properties such that all distances along specific edges are rational numbers. (This is equivalent to having all edge lengths be integers, since the denominators of rational numbers can be cleared by multiplication.)

A cuboid whose edges and face diagonals are integers is called an Euler brick. It is not known if there exists a point in a unit square all of whose distances from the corners are rational, although J. H. Conway and M. Guy found an infinite numbers of solutions to the problem of three such distances being integers, which involves solving

where , , and  are the three distances and  is the side length of the square (Guy 1994, p. 181). There are infinitely many solutions of the corresponding problem of integer distances from the corners of an equilateral triangle (Guy 1994, p. 183).

In 2001, E. Pegg found a small scalene triangle with side lengths 8, 22, and 19 possessing an internal point with distances 17, 6, and 4 from the respective vertices. This is equivalent to finding an integers-only solution of the equation for the six distances between vertices of a quadrilateral.

REFERENCES:

Guy, R. K. "Rational Distances from the Corner of a Square." §D19 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 181-185, 1994.