In geometry, the Braikenridge–Maclaurin theorem, named for 18th century British mathematicians William Braikenridge and Colin Maclaurin,[1] is the converse to Pascal's theorem. It states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line L, then the six vertices of the hexagon lie on a conic C; the conic may be degenerate, as in Pappus's theorem.[2]
Elliptic case
Hyperbolic case
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 six points on a conic (the vertices of a hexagon), the lines defined by opposite sides intersect in three collinear points. This can be reversed to construct the possible locations for a sixth point, given five existing ones.
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, JSTOR 531819
braikenridge, maclaurin, theorem, geometry, named, 18th, century, british, mathematicians, william, braikenridge, colin, maclaurin, converse, pascal, theorem, states, that, three, intersection, points, three, pairs, lines, through, opposite, sides, hexagon, li. In geometry the Braikenridge Maclaurin theorem named for 18th century British mathematicians William Braikenridge and Colin Maclaurin 1 is the converse to Pascal s theorem It states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line L then the six vertices of the hexagon lie on a conic C the conic may be degenerate as in Pappus s theorem 2 Elliptic case Hyperbolic case 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 six points on a conic the vertices of a hexagon the lines defined by opposite sides intersect in three collinear points This can be reversed to construct the possible locations for a sixth point given five existing ones References Edit 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 JSTOR 531819 Coxeter H S M Greitzer S L 1967 Geometry Revisited Washington DC Mathematical Association of America p 76 This elementary geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Braikenridge Maclaurin theorem amp oldid 986741503, wikipedia, wiki, book, books, library,
