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Send one of the points to infinity by a Mobius tranformation. Your set of points now has the property that the intersection of any two lines passing from pairs of points in the set is also in the set. Such configurations are either all collinear, dense in the plane, or one of these two exceptional cases:

• A point together with a collection of points in a line
• The four vertices of a convex quadrilateral

To prove your result we need to rule out the first case. This can be done by considering circumcircles of triangles with disjoint bases and the common apex. The result I used above is proved in "A dense planar point set from iterated line intersections" by D. Ismailescu and R. Radoicic.