Courant described the outline of an elementary proof of the Jordan curve theorem for polygons using the order of points:
The order of a point $p_0$ is defined by the net number of complete revolutions made by an arrow joining the point $p_0$ to an moving point $p$ as $p$ traverses the polygon $P$ once. Class $A$ is defined as the points not on $P$ with even order with respect to $P$. Class $B$ is defined as the points not on $P$ with odd order with respect to $P$.
How do you complete the proof that the polygon $P$ divides the points of the plane into two distinct domains using the above definitions?