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Topic: How Many Triangles are Created by n Lines in the Plane???
Replies: 0

 Daniel A. Asimov Posts: 35 Registered: 12/6/04
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 n-2 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