Points at Rational Distance from the Vertices of a Square

Abstract

Guy asks if there exists a point in the plane at rational distance to the corners of the unit square. Also known as the four-distance problem, we establish the equivalence of the problem to the existence of nontrivial solutions to a particular Pythagorean triple, from which we derive known conditions and establish new results. We then provide a generalization given by Barbara of the four-distance problem to regular polygons of unit side, in which a negative answer is almost always obtained.