Double Balanced Quadrilaterals

Abstract

The k-centroids G_k of a polygon, for k = 0, 1, 2, are the centroids of the polygon when the mass is equally distributed respectively between the vertices, along the perimeter, or across the area. A fundamental theorem by Al-Sharif, Hajja, and Krasopoulos in [1] asserts that the quadrilaterals with either G₀ = G₁ or G₀ = G₂ are precisely all parallelograms. Our main result describes the non-parallelograms with G₁ = G₂ by providing formulas for their diagonals in terms of the sides, as well as formulas for the ratios determined on the diagonals by their intersection point. In this way, we complete a fifteen-year-old problem by these three authors on characterizing all double balanced quadrilaterals. As an application, we show how our main theorem can be used to deduce their characterizations of double balanced circumscribed and cyclic quadrilaterals.

MSC:

• 51M04

Acknowledgment

The authors wish to thank the anonymous reviewers for their careful reading and suggestions. We thank Monthly’s Editor, Della Dumbaugh, and the Editorial Board for their constant support and encouragement throughout the reviewing process, and especially for their generous feedback and great insight that really helped make these results shine.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Allan Berele

Allan Berele earned his bachelors degree at the University of Rochester and his doctorate at the University of Chicago, under I. N. Herstein. Most of his research has been in rings with polynomial identities, but in recent years he has been focusing, together with S. Catoiu, on questions of how convex set theory relates to Euclidean geometry.

Department of Mathematics, DePaul University, Chicago, IL 60614

[email protected]

Stefan Catoiu

Stefan Catoiu received his Master’s degree from the University of Bucharest, and Ph.D. from the University of Wisconsin-Madison, under D. S. Passman. He then held a two-year visiting position at Temple University before joining the faculty at DePaul in 1999. His research is in algebra, real analysis, and geometry.

Department of Mathematics, DePaul University, Chicago, IL 60614

[email protected]