  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.
