S is an infinite set of points in the plane. The distance between any two points of S is integral. Prove that S is a subset of a straight line.
Solution
Suppose not. Take three non-collinear points A, B, C. Suppose AB = n. Any point P has PA + AB ≥ PB, so PB - PA ≤ n. Similarly PB - PA ≥ -n. But PB - PA is integral, so it must take one of the values -n, -(n-1), ... , 0, 1, ... , n. So P must lie on one of a finite number of hyperbolae |PB - PA| = k (regarding the allowed line pair as a degenerate hyperbola). Similarly it must lie on one of a finite number of hyperbolae |PA - PC| = h. But each pair of hyperbolae intersect in at most 4 points, so the number of points in addition to A, B, C is finite. Contradiction.
© John Scholes
jscholes@kalva.demon.co.uk
25 Feb 2002