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 $a,b,\text{and}c$, Newton arrived at the cubic equation $d^3-(a^2+b^2+c^2)d-2a\mathit{bc}=0$, 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.

MSC:

• Primary 11D25

Additional information

Notes on contributors

Mowaffaq Hajja

MOWAFFAQ HAJJA graduated from Al-Fadhiliyya High School in Toulkarm, Palestine. He earned his B.Sc. from the Middle East Technical University, Turkey, in 1972, and his Ph.D. from Purdue University, USA, in 1978, where he wrote his thesis in field theory under the supervision of T. T. Moh. After two years at Michigan State University, USA, he moved to Yarmouk University in Irbid, Jordan, where he has spent most of his time since then. He enjoys doing mathematics, where his main interests include algebra, geometry, means, and writing problems for the MAA journals and similar publications.

Jonathan Sondow

JONATHAN SONDOW graduated from Stuyvesant High School in New York City and the University of Wisconsin at Madison. In 1965 at the age of 22 he received a Ph.D. from Princeton University, where he wrote a thesis in differential topology supervised by John Milnor. After two years as a postdoc in Paris, Sondow held numerous academic posts in the U.S. and worked at the Institute for Defense Analyses. Currently he maintains a homepage at home.earthlink.net/∼jsondow/devoted to research in number theory and geometry. His Erdős number is two. On 10 March 2011, after a three-week visit to Keio University in Tokyo, he and his wife Hisayo returned home to Manhattan hours before the Great East Japan Earthquake.