MathDB

Let

P_{1}, \ldots P_{n}

be the sequence of vertices of a closed polygons whose sides may properly intersect each other at points other than the vertices. The external angle at

P_{i}

is defined as

180^\circ

minus the angle of rotation about

P_{i}

required to bring the ray

P_{i}P_{i-1}

onto the ray

P_{i}P_{i+1}

, taken in the range (

0^\circ, 360^\circ

). (Here

P_{0}=P_{n}

and

P_{1}=P_{n+1}

). Prove that if the sum of the external angles is a multiple of

720^\circ

, then the number of self-intersections is odd.

Problem Statement