On the Dual and Sharpened Dual of Sylvester's Theorem in the Plane

Given a collection of n lines in the Euclidean plane, not all coincident and not all parallel, we prove that there must be a point where precisely two lines intersect. We show that this result is a sharpened version of the dual of the classical Theorem of Sylvester. We consider Sylvester's Theorem as well as its various duals in hyperbolic space. Finally we obtain lower bounds on the number of ordinary points in arrangements satisfying the hypotheses of our sharpened dual theorem.

By: Jonathan Lenchner

Published in: RC23411 in 2004


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