In geometry, the Braikenridge–Maclaurin theorem, named for 18th century British mathematicians William Braikenridge and Colin Maclaurin (Mills 1984), is the converse to Pascal's theorem. It states that if the 3 intersection points of the lines through three sides of a hexagon lie on a line, then the 6 vertices of the hexagon lie on a conic; the conic may be degenerate, as in Pappus's theorem. (Coxeter & Greitzer 1967, p. 76). The Braikenridge–Maclaurin theorem may be applied in the Braikenridge–Maclaurin construction, which is a synthetic construction of the conic defined by five points, by varying the sixth point. Namely, Pascal's theorem states that given 6 points on a conic (a hexagon), the lines defined by opposite sides intersect in three collinear points. This can be reversed to construct the possible locations for a 6th point, given 5 existing ones.

The theorem was generalized by Möbius in 1847, as follows: suppose a polygon with 4n + 2 sides is inscribed in a conic section, and opposite pairs of sides are extended until they meet in 2n + 1 points. Then if 2n of those points lie on a common line, the last point will be on that line, too.

References

* Mills, Stella (March 1984), "Note on the Braikenridge-Maclaurin Theorem", Notes and Records of the Royal Society of London (The Royal Society) 38 (2): 235–240, doi:10.1098/rsnr.1984.0014, http://www.jstor.org/stable/531819
* Coxeter, H. S. M.; Greitzer, S. L. (1967), Geometry Revisited, Washington, DC: Mathematical Association of America, p. 76