Editing Pascal's theorem (section)

== Related results ==

Pascal's theorem is the [[polar reciprocation|polar reciprocal]] and [[projective dual]] of [[Brianchon's theorem]]. It was formulated by [[Blaise Pascal]] in a note written in 1639 when he was 16 years old and published the following year as a [[Broadside (printing)|broadside]] titled "Essay pour les coniques. Par B. P."<ref name=orig>{{harvnb|Pascal|1640}}, translation {{harvnb|Smith|1959|p=326}}</ref>

Pascal's theorem is a special case of the [[Cayley–Bacharach theorem]].

A degenerate case of Pascal's theorem (four points) is interesting; given points {{math|''ABCD''}} on a conic {{math|Γ}}, the intersection of alternate sides, {{math|''AB'' ∩ ''CD''}}, {{math|''BC'' ∩ ''DA''}}, together with the intersection of tangents at opposite vertices {{math|(''A'', ''C'')}} and {{math|(''B'', ''D'')}} are collinear in four points; the tangents being degenerate 'sides', taken at two possible positions on the 'hexagon' and the corresponding Pascal line sharing either degenerate intersection. This can be proven independently using a property of [[pole and polar|pole-polar]]. If the conic is a circle, then another degenerate case says that for a triangle, the three points that appear as the intersection of a side line with the corresponding side line of the [[Gergonne triangle]], are collinear.

Six is the minimum number of points on a conic about which special statements can be made, as [[five points determine a conic]].

The converse is the [[Braikenridge–Maclaurin theorem]], named for 18th-century British mathematicians [[William Braikenridge]] and [[Colin Maclaurin]] {{harv|Mills|1984}}, which states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line, then the six vertices of the hexagon lie on a conic; the conic may be degenerate, as in Pappus's theorem.<ref>{{harvs|txt|last1=Coxeter|first1 = H. S. M. | authorlink1 = Harold Scott MacDonald Coxeter | last2 = Greitzer | first2 = Samuel L. | authorlink2 = Samuel L. Greitzer| |year=1967|p=76}}</ref> 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.

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