View source for Braikenridge–Maclaurin theorem

{{short description|Converse to Pascal's theorem In geometry}}
[[File:Braikenridge–Maclaurin theorem.svg|thumb|Elliptic case]]
[[File:Braikenridge–Maclaurin theorem 2.svg|thumb|Hyperbolic case]]
In [[Geometry|geometry]], the '''{{visible anchor|Braikenridge–Maclaurin theorem}}''', named for 18th century British mathematicians William Braikenridge and Colin Maclaurin,{{r|mills}} 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.{{r|cg}}

The Braikenridge–Maclaurin theorem may be applied in the Braikenridge–Maclaurin construction, which is a [[Synthetic geometry|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==
{{reflist|refs=

<ref name=mills>{{ Citation | title = Note on the Braikenridge-Maclaurin Theorem | first = Stella | last = Mills | journal = Notes and Records of the Royal Society of London | volume = 38 | number = 2 |date=March 1984 | pages = 235–240 | publisher = The Royal Society | jstor = 531819 | doi = 10.1098/rsnr.1984.0014 }}</ref>

<ref name=cg>{{ Citation | last1 = Coxeter | first1 = H. S. M. |  last2 = Greitzer | first2 = S. L. |  title = Geometry Revisited | location = Washington, DC | publisher = Mathematical Association of America | page = 76 | year = 1967}}</ref>

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[[Category:Conic sections]]



{{Sourceattribution|Braikenridge–Maclaurin theorem}}