Maximum number of triangles given a number of lines.

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This is just something I was wondering in my spare time.

The problem: What is the maximum amount of triangles you can create given a fixed number of lines. The lines are of infinite length and can be placed anywhere you want. The triangles must not overlap though (so the star of david for example would consist only of 6 triangles and not 8)

Example:

3 lines -> max 1 triangle

4 lines -> max 2 triangles

Does anybody have any idea how to tackle this problem? Has anybody done any work on something similar?