Assume you have a set X (of points) and subsets $A_i$ with the following conditions:

(1): For any two points in X exactly one of the sets contain them. (2): Any two subsets intersect at most at one point. (3): One of the subsets is of size 2 exactly.

The question is: Can we find $|X|$ points in the plane such that the $A_i$ are exactly the lines through them.

(3) is exactly Sylvester's theorem, I am wondering if that is the only obstruction.