If n is the number of lines then the number of intersections is p=½n(n-1).
The reason is that each new line intersects the existing lines once only but the existing lines already intersect, so the number of those intersections has to be added to the set of new ones.
If we have P intersection points already from n lines then when we add the (n+1)th we have n new intersections to add so we have P+n intersections. We can write this as p[n+1]=p[n]+n.
When n=2, p=1 by the formula. That’s the base case.
Assume the formula is true for all n then p[n+1]=½n(n-1)+n=½n²-½n+n=½n²+½n=½(n+1)n=p[n+1]. That’s proof by induction.