![Braikenridge–Maclaurin theorem 2](https://upload.wikimedia.org/wikipedia/commons/4/49/Braikenridge%E2%80%93Maclaurin_theorem_2.svg)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the , named for 18th century British mathematicians
William Braikenridge
William Braikenridge (also Brakenridge) (c.1700–1762) was a Scottish mathematician and cleric, a Fellow of the Royal Society from 1752.
Life
He was son of John Braikenridge of Glasgow. s:Page:Alumni Oxoniensis (1715–1886) volume 1.djvu/169 In ...
and
Colin Maclaurin
Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for bei ...
, is the converse to
Pascal's theorem
In projective geometry, Pascal's theorem (also known as the ''hexagrammum mysticum theorem'') states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by ...
. 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.
The Braikenridge–Maclaurin theorem may be applied in the
Braikenridge–Maclaurin construction
In Euclidean and projective geometry, just as two (distinct) points determine a line (a degree-1 plane curve), five points determine a conic (a degree-2 plane curve). There are additional subtleties for conics that do not exist for lines, and thu ...
, 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
Theorems about polygons
Conic sections
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