MathDB

Suppose that a closed oriented polygonal line

\mathcal{L}

in the plane does not pass through a point

O

, and is symmetric with respect to

O

. Prove that the winding number of

\mathcal{L}

around

O

is odd.
The winding number of

\mathcal{L}

around

O

is defined to be the following sum of the oriented angles divided by

2\pi

\deg_O\mathcal{L} := \dfrac{\angle A_1OA_2+\angle A_2OA_3+\dots+\angle A_{n-1}OA_n+\angle A_nOA_1}{2\pi}.

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