A characterization of the convexity of cyclic polygons in terms of the central angles
Let P be a cyclic n-gon with n ≥ 3, the central angles θ 0 (-π, π], ... , θ n-1 (-π,π], and the winding number w := (θ 0 +...+ θ n-1)/(2π). 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 θ 0,..., θ n-1 > 0; (II) w = -1 and θ 0,..., θ n-1 < 0; (III) w = 0 and exactly one of the angles θ 0,...,θ n-1 is negative; (IV) w = 0 and exactly one of the angles θ 0,..., θ n-1 is positive. © 2007 Birkhaueser.
Journal of Geometry
A characterization of the convexity of cyclic polygons in terms of the central angles.
Journal of Geometry,
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