Braikenridge-Maclaurin theorem

The Braikenridge-Maclaurin theorem is a result in plane geometry, named for the eighteenth century British mathematicians William Braikenridge and Colin Maclaurin. It is the converse of Pascal's theorem. It states that if the three points where the three pairs of lines through opposite sides of a hexagon meet all lie on one line, then the six vertices of the hexagon lie on a conic; the conic may be degenerate, as in Pappus's hexagon theorem. The theorem underlies the Braikenridge-Maclaurin construction, a synthetic way to build the conic defined by five points by varying the sixth point: Pascal's theorem says that the lines through opposite sides of a hexagon inscribed in a conic meet in three collinear points, and reversing this gives the possible positions of a sixth point when five points are already given.