1) Let L be a line in coordinate 2- space R^2 or 3- space R^3, let x be a point not on L, and let p1,...pn be points on L. Prove that the lines xp1,...xpn are distinct. Why does this imply that R^2 and R^3 contain infinitely many lines?
2) Suppose that L1,...Ln are lines in coordinate 2- space R^2 or 3- space R^3. Prove that there is a point q which does not lie on any of these lines. (hint: Take a line M which is different from each of L1,...Ln; for each j we know that M and Lj have at most one point in common, but we also know how M has infinitely many points.)
Thanks a whole lot.