info prev up next book cdrom email home

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.

{\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

{\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.

In general, it is also true that

{\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}.

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


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.

© 1996-9 Eric W. Weisstein