A characterization of the convexity of cyclic polygons in terms of the central angles

Abstract.

Let ${\mathcal{P}}$ 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 ${\mathcal{P}}$ are assumed to be all distinct from one another. It is then proved that ${\mathcal{P}}$ 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.