# How to count the additional lines?

A plane has 6 lines of which no two lines are parallel and no three are concurrent. Their points of intersection are joined, how many of additional lines are so formed?

I know that number of points of intersection for $n$ lines would be $\sum \limits_{i=1}^{n-1} i=\frac{n(n-1)}{2}$, but then how do I do the rest?

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Are you sure that it doesn’t ask for the maximum possible number of additional lines? –  Brian M. Scott Nov 4 '11 at 22:35
What if three intersection points lie on a line? Do we call the resulting number of lines 1, 2, or 3? –  mixedmath Nov 4 '11 at 22:38