We bring together several new results related to the classical Sylvester-Gallai Theorem and its recently noted sharp dual. In 1951 Dirac and Motzkin conjectured that a configuration of n not all collinear points must admit at least n/2 ordinary connecting lines. There are two known counterexamples, when n=7 and n=13. We provide a construction that yields both counterexamples and note that the common construction cannot be extended to provide additional counterexamples. We give examples to show that the Sylvester-Gallai Theorem and its sharp dual are both false on the Torus.
By: Hervé Brönnimann, Jonathan Lenchner
Published in: RC23619 in 2005
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