Kelly proved that "ℓ" is ordinary, by contradiction. Assume that "ℓ" is not ordinary. Then it goes through at least three points of "S". At least two of these are on the same side of "P'", the perpendicular projection of "P" on "ℓ". Call them "B" and "C", with "B" being closest to "P'" (and possibly coinciding with it")". Draw the connecting line "m" passing through "P" and "C," and the perpendicular from "B" to "B'" on "m" . Then "BB'" is shorter than "PP"'. This follows from the fact that "PP'C" and "BB'C" are similar triangles, one contained inside the another. However, this contradicts the original definition of "P" and "ℓ" as the point-line pair with the smallest positive distance. So the assumption that "ℓ" is not ordinary cannot be true, QED. In 1941 (thus, prior to Erdős publishing the question and Gallai's subsequent proof) Melchior showed that any nontrivial finite arrangement of lines in the projective plane has at least three ordinary points. By duality, this results also says that any finite nontrivial set of points on the plane has at least three ordinary lines. Melchior observed that, for any graph embedded in the real projective plane, the formula formula_3 must equal formula_4, the Euler characteristic of the projective plane. Here formula_5, formula_6, and formula_7 are the number of vertices, edges, and faces of the graph, respectively. Any nontrivial line arrangement on the projective plane defines a graph in which each face is bounded by at least three edges, and each edge bounds two faces; so, double counting gives the additional inequality formula_8. Using this inequality to eliminate formula_7 from the Euler characteristic leads to the inequality formula_10. But if every vertex in the arrangement were the crossing point of three or more lines, then the total number of edges would be at least formula_11, contradicting this inequality. Therefore, some vertices must be the crossing point of only two lines, and as Melchior's more careful analysis shows, at least three ordinary vertices are needed in order to satisfy the inequality formula_10. As note, the same argument for the existence of an ordinary vertex was also given in 1944 by Norman Steenrod, who explicitly applied it to the dual ordinary line problem.