to find the closest given point to each line through two given points, in time , was given earlier by , as a subroutine for finding the minimum-area triangle determined by three of a given set of points. The same paper of also shows how to construct the dual arrangement of lines to the given points (as used in Melchior and Steenrod's proof) in the same time, , from which it is possible to identify all ordinary vertices and all ordinary lines. first showed how to find a single ordinary line (not necessarily the one from Kelly's proof) in time , and a simpler algorithm with the same time bound was described by . The algorithm of is based on Coxeter's proof using ordered geometry. It performs the following steps: Choose a point that is a vertex of the convex hull of the given points. Construct a line that passes through and otherwise stays outside of the convex hull. Sort the other given points by the angle they make with , grouping together points that form the same angle. If any of the points is alone in its group, then return the ordinary line through that point and . For each two consecutive groups of points, in the sorted sequence by their angles, form two lines, each of which passes through the closest point to in one group and the farthest point from in the other group. For each line in the set of lines formed in this way, find the intersection point of with Return the line whose intersection point with is the closest to . As the authors prove, the line returned by this algorithm must be ordinary. The proof is either by construction if it is returned by step 4, or by contradiction if it is returned by step 7: if the line returned in step 7 were not ordinary, then the authors prove that there would exist an ordinary line between one of its points and , but this line should have already been found and returned in step 4. The number of ordinary lines While the Sylvester–Gallai theorem states that an arrangement of points, not all collinear, must determine an ordinary line, it does not say how many must be determined. Let be the minimum number of ordinary lines determined over every set of non-collinear points. Melchior's proof showed that . raised the question of