Self-Transversality Theorem

Let $j$, $r$, and $s$ be distinct Integers (mod $n$), and let $W_i$ be the point of intersection of the side or diagonal $V_iV_{i+j}$ of the $n$-gon $P=[V_1, \ldots, V_n]$ with the transversal $V_{i+r}V_{i+s}$. Then a Necessary and Sufficient condition for

$$\prod_{i=1}^n \left[{V_iW_i\over W_iV_{i+j}}\right]=(-1)^n,$$

where $AB\vert\vert CD$ and

$$\left[{AB\over CD}\right],$$

is the ratio of the lengths $[A, B]$ and $[C, D]$ with a plus or minus sign depending on whether these segments have the same or opposite direction, is that

1. $n=2m$ is Even with $j\equiv m\ \left({{\rm mod\ } {n}}\right)$ and $s\equiv r+m\ \left({{\rm mod\ } {n}}\right)$,

2. $n$ is arbitrary and either $s\equiv 2r$ and $j\equiv 3r$, or

3. $r\equiv 2s\ \left({{\rm mod\ } {n}}\right)$ and $j\equiv 3s\ \left({{\rm mod\ } {n}}\right)$.

References

Grünbaum, B. and Shepard, G. C. ``Ceva, Menelaus, and the Area Principle.'' Math. Mag. 68, 254-268, 1995.