A Characterization of the Convexity of Cyclic Polygons in Terms of the   Central Angles
  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. 
Iosif Pinelis