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How Many Triangles are Created by n Lines in the Plane???
Posted:
Jan 17, 1995 3:52 PM


Suppose we are given n lines in the plane in "general position", which in the present case we define to mean the following:
A. no 2 lines are parallel or identical B. no 3 lines have common intersection C. no 3 of their intersection points are collinear unless they all lie on one of the n lines.
PROBLEM: Prove that among the regions created by the n lines, there are at least n2 triangles.
(Thanks to Mark Manasse for communicating this problem to me, which he reports was found in a school textbook by a friend of his.)
Notes: 1. I don't know the solution of this problem. 2. "General position" as defined above should be sufficient to solve this problem, but a more relaxed version (probably omitting condition C) may also suffice.
Dan Asimov



