Hoehn's Theorem

$\begin{figure}\begin{center}\BoxedEPSF{HoehnsTheorem.epsf scaled 700}\end{center}\end{figure}$

A geometric theorem related to the Pentagram and also called the Pratt-Kasapi Theorem.

$\begin{displaymath} {\vert V_1W_1\vert\over \vert W_2V_3\vert}{\vert V_2W_2\vert... ...\vert W_5V_1\vert}{\vert V_5W_5\vert\over \vert W_1V_2\vert}=1 \end{displaymath}$

$\begin{displaymath} {\vert V_1W_2\vert\over \vert W_1V_3\vert}{\vert V_2W_3\vert... ...vert W_4V_1\vert}{\vert V_5W_1\vert\over \vert W_5V_2\vert}=1. \end{displaymath}$

In general, it is also true that

$\begin{displaymath} {\vert V_iW_i\vert\over\vert W_{i+1}V_{i+2}\vert}={\vert V_i... ...i+1}V_{i+2}V_{i+3}\vert\over\vert V_{i+2}V_{i+3}V_{i+1}\vert}. \end{displaymath}$

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

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, 254-268, 1995.

Pinkernell, G. M. ``Identities on Point-Line Figures in the Euclidean Plane.'' Math. Mag. 69, 377-383, 1996.