Four lines can intersect in at most $\frac{4^{2}-4}{2} = 6$ points. And in fact you can find an example of lines intersecting in 0, 1, 3, 4, 5 and 6 points. All but 2. Obviously there isn't any way how four lines can intersect in two points. But how to prove it?
[Math] Can 4 lines intersect in 2 points
geometryrecreational-mathematics
Best Answer
I prove this by assuming such a construction exists and deriving a contradiction.
Let's pick our two points of intersection for the construction and go from there. Since each has at least two lines passing through it, let us consider one of the lines ($L_1$) going through one point and two of the lines ($L_2, L_3$) going through the other. (If one of the lines happens to be passing through both points, we just pick the other three for the observation.)
Since neither of $L_2$ and $L_3$ intersect with $L_1$, they are both parallel to $L_1$ as well as each other. But since $L_2$ and $L_3$ pass through a common point, they must coincide, thus contradicting the hypothesis that there are 4 unique lines.