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A Versatile Theorem: What is it?
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Copyright © 1996-2008 Alexander Bogomolny
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.
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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.
Copyright © 1996-2008 Alexander Bogomolny
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