Kelly's proof of the existence of an ordinary line can be turned into an algorithm that finds an ordinary line by searching for the closest pair of a point and a line through two other points. report the time for this closest-pair search as formula_49, based on a brute-force search of all triples of points, but an algorithm to find the closest given point to each line through two given points, in time formula_50, 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, formula_50, 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 formula_2, and a simpler algorithm with the same time bound was described by .