A Characterization of the Convexity of Cyclic Polygons in Terms of the Central Angles

Abstract

Let P be a cyclic n-gon with n\ge3, the central angles \th_0,...,\th_{n-1} in (-\pi,\pi], and the winding number w:=(\th_0+...+\th_{n-1})/(2\pi). The vertices of P are assumed to be all distinct from one another. It is then proved that P is convex if and only if one of the following four conditions holds: (I) w=1 and \th_0,...,\th_{n-1}>0; (II) w=-1 and \th_0,...,\th_{n-1}<0; (III) w=0 and exactly one of the angles \th_0,...,\th_{n-1} is negative; (IV) w=0 and exactly one of the angles \th_0,...,\th_{n-1} is positive.