Dual to Pappus' Theorem: What is it?
A Mathematical Droodle

A few words

Copyright © 1996-2009 Alexander Bogomolny

The applet appears to demonstrate the following fact. Three vertical and three horizontal straight lines intersect at 9 points. If the pairs of points are joint by straight lines, then there are serveral triplets of concurrent lines. For example, denote the vertical lines by the capital letters A, B, C, and the horizontal lines by the lower case letters: a, b, c. Let, for two lines a and b, a·b stand for their point of intersection. Let's also agree that, for two points P and Q, P·Q is used interchangeably with PQ to denote the line incident to the two of them. Then the applet suggests that the lines Aa·Cb, Ba·Cc, and Ca·Bb are concurrent, as are the lines Ab·Ba, Aa·Cc, and Bc·Cb, and there are more of such concurrencies.

It's straightforeward to prove that using analytic geometry. But, perhaps surprisingly, the statement has a meaning in projective geometry. Furthermore, it's in fact a stylized reformulation of the dual to Pappus theorem. Indeed, that dual theorem asserts that for two triplets of concurrent lines, the joints of some of their intersections concur. In projective geometry, parallel lines concur at a point at infinity. So indeed in the applet we have two triplets of concurrent lines and its easy to verify that the joints are the same in both theorems. (The multitude of choices stems from the different orderings of the lines in the triplets.)

Furthermore, in terms of projective geometry the "general" theorem and the one illustrated by the applet are equivalent, for it's possible to projectively transform a "general" configuration into the one of two sets of parallel lines with the lines from two different triplets perpendicular. (First project one of the points to infinity. Then, before following suite with the other point, rotate the plane so that the lines in the first triplet become parallel to the horizon.)

Note that via the Principle of Duality we obtain an additional proof of Pappus' theorem itself.

The situation is curious: a result in projective geometry that knows not angles nor distances is obtained by means of analytic geometry in which the notions of angle and distance play a central role via the similar situation in the discussion on isogonal concurrencies.)

References

1. D. Wells, The Penguin Dictionary of metry, Penguin Books, 1991

Copyright © 1996-2009 Alexander Bogomolny