Carnot's Polygon Theorem

If $P_1$, $P_2$, ..., are the Vertices of a finite Polygon with no ``minimal sides'' and the side $P_iP_j$ meets a curve in the Points $P_{ij1}$ and $P_{ij2}$, then

$${\prod_i \overline{P_1P_{12i}}\prod_i \overline{P_2P_{23i}}\... ... \overline{P_NP_{N1i}}\cdots\prod_i \overline{P_2P_{2i1}}} =1,$$

where $\overline{AB}$ denotes the Distance from Point $A$ to $B$.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 190, 1959.