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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,...,V_n] with the transversal V_(i+r)V_(i+s). Then a necessary and sufficient condition for

 product_(i=1)^n[(V_iW_i)/(W_iV_(i+j))]=(-1)^n,

where AB∥CD and

 [(AB)/(CD)],

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=m (mod n) and s=r+m (mod n),

2. n is arbitrary and either s=2r and j=3r, or

3. r=2s (mod n) and j=3s (mod n).


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References

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

Referenced on Wolfram|Alpha

Self-Transversality Theorem

Cite this as:

Weisstein, Eric W. "Self-Transversality Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Self-TransversalityTheorem.html

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