Newton Quadrilaterals, the Associated Cubic Equations, and Their Rational Solutions

Abstract

A Newton quadrilateral is a convex quadrilateral inscribed in a circle, with one side a diameter. For diameter d and the other side lengths , Newton arrived at the cubic equation , which we shall refer to as Newton’s equation. Its positive integer solutions were found by P. Bachmann. All (positive and nonpositive) integer solutions were found by A. Oppenheim in terms of logarithmic and hyperbolic functions. A formula that produces infinitely many rational solutions was found by N. Anning, using an elegant relation among the cosines of the angles of a triangle. All rational solutions of Newton’s equation were found, using a short and elegant method based on linear algebra, by I. A. Barnett. The same results of Barnett were recently obtained independently by M. Hajja, using a relation among the cosines of the angles of what he called a generalized triangle. In the present paper, we use a direct algebraic method that also yields all rational solutions, albeit parameterized in a way very different from that obtained by Barnett (and Hajja), and we establish an explicit, nonobvious, birational correspondence between the two parameterizations. We also treat the nonconvex version of Newton’s problem and several related issues.