The Largest Quadrilateral is Cyclic: A New Geometric Proof

Abstract

Although 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 [5] and [11]. 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.