Self-Transversality Theorem

Let , , and be distinct integers (mod ), and let be the point of intersection of the side or diagonal of the -gon with the transversal . Then a necessary and sufficient condition for

where and

is the ratio of the lengths and with a plus or minus sign depending on whether these segments have the same or opposite direction, is that

1. is even with and ,

2. is arbitrary and either and , or

3. and .