# What's the most triangles you can make with 4, 5 or 6 straight lines?

All the triangles can stick together. The triangles counted is the independent triangles, triangles made up of two shapes, a triangle made up from 3 shapes, or the outline of the shape consisting of 4, 5 or 6 lines.

• In Euclidean geometry? – Peter Taylor Jan 6 at 9:11

I seem to recall that the solution to this problem for $$n$$ lines is

$$n \choose 3$$, so for 4 lines it's 4, for 5 lines it's 10 and for 6 lines it's 20.

The idea is that

each combination of 3 lines generates exactly one triangle. They intersect in at most 3 points, which form the triangle. Actually, as long as you don't choose parallel lines, they will intersect in exactly 3 points. There are $$n \choose 3$$ of these combinations.

For example, here is the solution for 6 lines:

The triangles are formed by: