1. The problem statement, all variables and given/known data edit: WHOOPS. Almost forgot to 3. The attempt at a solution Ok. So I began with the line l_1 with exactly 6 points on it, A, B, C, D, E, F. (Axiom 2). Now, by axiom 1, there must exist 2 additional points not on this line. So I formed those, G and H. No by Axiom 3, there exists a line between each of the points on L_1 and G and H, as well as a line between G and H. I drew those. Notice, however, that the line HF is parallel to GA, GB, GC, GD, and GE. This contradicts axiom 4 which states that through G, only 1 line is parallel to HF!