line can be found as a line of slope closest to zero; for details, see . The 1941 proof by Melchior uses projective duality to convert the problem into an equivalent question about arrangements of lines, which can be answered using Euler's polyhedral formula. Another proof by Leroy Milton Kelly shows by contradiction that the connecting line with the smallest nonzero distance to another point must be ordinary. And, following an earlier proof by Steinberg, H. S. M. Coxeter showed that the metric concepts of slope and distance appearing in Gallai's and Kelly's proofs are unnecessarily powerful, instead proving the theorem using only the axioms of ordered geometry. Kelly's proof This proof is by Leroy Milton Kelly. call it "simply the best" of the many proofs of this theorem. Suppose that a finite set of points is not all collinear. Define a connecting line to be a line that contains at least two points in the collection. By finiteness, must have a point and a connecting line that are a positive distance apart but are closer than all other point-line pairs. Kelly proved that is ordinary, by contradiction. Assume that is not ordinary. Then it goes through at least three points of . At least two of these are on the same side of , the perpendicular projection of on . Call them and , with being closest to (and possibly coinciding with it). Draw the connecting line passing through and , and the perpendicular from to on . Then is shorter than . This follows from the fact that and are similar triangles, one contained inside the another. However, this contradicts the original definition of and as the point-line pair with the smallest positive distance. So the assumption that is not ordinary cannot be true, QED. Melchior's proof 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 must equal , the Euler characteristic of the projective plane. Here , , and are the number of vertices, edges, and faces of the graph, respectively. Any nontrivial line arrangement on the projective