How Many Triangles are Created by n Lines in the Plane?

Question

Suppose we are given n lines in the plane in "general position", which in the present case we define to mean the following:

1. no 2 lines are parallel or identical
2. no 3 lines have common intersection
3. 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.

Answer 1

This problem known since 1889 and solved by Shannon in 1979. In 1992 a beautiful solution was given in "On a problem of combinatorial geometry" by Belov. If you read Russian check also his note in "Kvant" magazine. An other simple proof was given in "Triangles in Euclidean arrangements" by Felsner and Kriegel

Here is Belov's idea:

Assuming that number of triangles is less then $n-2$ then you can fix two lines and move the rest so that the perimeter of each triangle stay the same. Note that this process does not change number of triangles.

This operation can be used to separate bunches of lines and this way it reduces the problem to the one with smaller number of lines.