In Number Theory, Logic, and Cryptography

Rational Distance


Given a unit square, can you find any point in the same plane, either inside or outside the square, that is a rational distance from all four corners?

Or, put another way, given a square ABCD of any size, can you find a point P in the same plane such that the distances AB, PA, PB, PC, and PD are all integers?







The problem is to find such a point, or prove that no such point can exist.


For further information, please see:

[1] Guy, Richard K. Unsolved Problems in Number Theory, Vol. 1, Springer-Verlag, 2nd ed. 1991, 181-185.


[2] Barbara, Roy. "The rational distance problem", Mathematical Gazette 95, March 2011, 59-61.


You can check for contributions to this problem on the solutions page.



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