Four lines are drawn on a plane with no two parallel....

Question

I am stuck on the following problem:

Four lines are drawn on a plane with no two parallel and no three concurrent.Lines are drawn joining the points of intersection of the previous four lines. Number of new lines obtained this way is : $\,\,3,\,\, 5, \,\, 2,\,\,12.$

My Attempt:

I think the answer will be $2$ but the answer key says it will be $3$. Can someone explain where I went wrong?

Answer 1

Using the hints given by @AJStas, I can say that the number of new lines obtained this way will be $3.$