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Hoehn's Theorem

HoehnsTheorem

A geometric theorem related to the pentagram and also called the Pratt-kasapi theorem. It states

(|V_1W_1|)/(|W_2V_3|)(|V_2W_2|)/(|W_3V_4|)(|V_3W_3|)/(|W_4V_5|)(|V_4W_4|)/(|W_5V_1|)(|V_5W_5|)/(|W_1V_2|)=1
(1)
(|V_1W_2|)/(|W_1V_3|)(|V_2W_3|)/(|W_2V_4|)(|V_3W_4|)/(|W_3V_5|)(|V_4W_5|)/(|W_4V_1|)(|V_5W_1|)/(|W_5V_2|)=1.
(2)

In general, it is also true that

(|V_iW_i|)/(|W_(i+1)V_(i+2)|)=(|V_iV_(i+1)V_(i+4)|)/(|V_iV_(i+1)V_(i+2)V_(i+4)|)(|V_iV_(i+1)V_(i+2)V_(i+3)|)/(|V_(i+2)V_(i+3)V_(i+1)|).
(3)

This type of identity was generalized to other figures in the plane and their duals by Pinkernell (1996).

See also

Ceva's Theorem, Menelaus' Theorem

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References

Chou, S. C. Mechanical Geometry Theorem Proving. Dordrecht, Netherlands: Reidel, 1987.Grünbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the Area Principle." Math. Mag. 68, 254-268, 1995.Hoehn, L. "A Menelaus-Type Theorem for the Pentagram." Math. Mag. 68, 121-123, 1995.Pinkernell, G. M. "Identities on Point-Line Figures in the Euclidean Plane." Math. Mag. 69, 377-383, 1996.

Referenced on Wolfram|Alpha

Hoehn's Theorem

Cite this as:

Weisstein, Eric W. "Hoehn's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HoehnsTheorem.html

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