A Versatile Theorem: What is it?
A Mathematical Droodle


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A Versatile Pappus' Theorem

The applet illustrates a theorem that can be used to establish the Minimax Theorem for two-person zero-sum games.

Assume the points E and F are arbitrarily selected on the sides AD and, respectively, BC of a rectangle ABCD. Let K be the point of intersection of AF and CE, and H that of BE and DF. Then HK||AB.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

The theorem is just a particular case of Pappus' theorem: if (there) Ab||aB, then, since Ab·aB, Bc·bC, and Ca·cA are collinear, the line through Bc·bC and Ca·cA is parallel to both Ab and aB. This is exactly the statement of the theorem at hand, where D and E replace B and C, while B, C, F replace a, b, and c.

From the association with Pappus' theorem, it is clear that ABCD need not be a rectangle. It may be any parallelogram: the conclusion HK||AB is still valid. In the most general setting, the three lines AB, CD, and HK meet in a finite point.

|Activities| |Contact| |Front page| |Contents| |Geometry| |Store|

Copyright © 1996-2017 Alexander Bogomolny

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