The Largest Quadrilateral is Cyclic: A New Geometric Proof

Abstract

AbstractAlthough treated as “obvious” since antiquity, the first complete proof that “a quadrilateral with given sides achieves the maximum area when it is cyclic” is attributed to Bretschneider (1842), who proved it using trigonometry. Peter (2003) proved it using calculus. It also follows from the isoperimetric inequality, proved geometrically in [Citation5] and [Citation11]. Here we give a new Euclidean geometric proof, starting from a different maximization problem: Find the tallest vertical line segment sandwiched between two semi-circles in a plane lying on opposite sides of a horizontal line with (partially) overlapping diameters.MSC: 51-01 AcknowledgmentI thank my colleague Professor Patrick Morton for permitting me to attend his Geometry for High School Teachers class and for reading an earlier draft of this paper. I am grateful to two referees for giving me many helpful suggestions.Additional informationNotes on contributorsJyotirmoy Sarkar Jyotirmoy Sarkar(jsarkar@iupui.edu) received his Ph.D. in Statistics from the University of Michigan, Ann Arbor. He is a professor and statistics consultant at Indiana University-Purdue University Indianapolis. His research interests are enumeration, probability, stochastic processes, and reliability theory. When he has time, he reads novels, cares for plants, and invents mathematical puzzles. ORCiD: 0000-0001-5002-5845