four lines. two points intersect. each point, except one, is the intersection of two lines.

Question

there ought to be a drawing of this

Answer 1

Start with 2 lines. There is only 1 intersection. Add another line. This will intersect the other 2 once each. But these lines have already intersected, so the number of intersections is 3. Add a 4th line. This will intersect each of the other 3, which have already intersected at 3 points. So the number of intersections increases to 6. Add a 5th line and the intersections increase by 4 to make 10. If we have n lines, then we add another, we increase the number of intersections by n for n+1 lines.

Thus we have an arithmetic series 0, 1, 3, 6, 10, 15. The nth term is n(n-1)/2. If n=35, we have 35*34/2=595 intersections.