Pascal's theorem: Difference between revisions

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[[Image:Pascal'sTheoremLetteredColored.PNG|thumb|250px|Self-crossing hexagon {{math|''ABCDEF''}}, inscribed in a circle. Its sides are extended so that pairs of opposite sides intersect on Pascal's line. Each pair of extended opposite sides has its own color: one red, one yellow, one blue. Pascal's line is shown in white.]]

In [[projective geometry]], '''Pascal's theorem''' (also known as the '''''hexagrammum mysticum theorem''''', [[Latin]] for mystical [[hexagram]]) states that if six arbitrary points are chosen on a [[conic section|conic]] (which may be an [[ellipse]], [[parabola]] or [[hyperbola]] in an appropriate [[affine plane]]) and joined by line segments in any order to form a [[hexagon]], then the three pairs of opposite [[Edge (geometry)|sides]] of the hexagon ([[extended side|extended]] if necessary) meet at three points which lie on a straight line, called the '''Pascal line''' of the hexagon. It is named after [[Blaise Pascal]].

The theorem is also valid in the [[Euclidean plane]], but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel.

This theorem is a generalization of [[Pappus's hexagon theorem|Pappus's (hexagon) theorem]], – Pappus's theoremwhich is the special case of a [[degenerate conic]] of two lines. with three points on each line.

== Euclidean variants ==

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==''Hexagrammum Mysticum''==

If six unordered points are given on a conic section, they can be connected into a hexagon in 60 different ways, resulting in 60 different instances of Pascal's theorem and 60 different Pascal lines. This [[projective configuration|configuration]] of 60 lines is called the ''Hexagrammum Mysticum''.<ref>{{harvnb|Young|1930|p=67}} with a reference to Veblen and Young, ''Projective Geometry'', vol. I, p. 138, Ex. 19.</ref><ref>{{harvnb|Conway|Ryba|2012}}</ref>

As [[Thomas Kirkman]] proved in 1849, these 60 lines can be associated with 60 points in such a way that each point is on three lines and each line contains three points. The 60 points formed in this way are now known as the '''Kirkman points'''.<ref>{{harvnb|Biggs|1981}}</ref> The Pascal lines also pass, three at a time, through 20 '''Steiner points'''. There are 20 '''Cayley lines''' which consist of a Steiner point and three Kirkman points. The Steiner points also lie, four at a time, on 15 '''Plücker lines'''. Furthermore, the 20 Cayley lines pass four at a time through 15 points known as the '''Salmon points'''.<ref>{{harvnb|Wells|1991|p=172}}</ref>

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== Degenerations of Pascal's theorem ==

[[File:Pascal-3456.png|450px|thumb|Pascal's theorem: degenerations]]

There exist 5-point, 4-point and 3-point degenerate cases of Pascal's theorem. In a degenerate case, two previously connected points of the figure will formally coincide and the connecting line becomes the tangent at the coalesced point. See the degenerate cases given in the added scheme and the external link on ''circle geometries''. If one chooses suitable lines of the Pascal-figures as lines at infinity one gets many interesting figures on [[parabola#Properties of a parabola related to Pascal's theorem|parabolas]] and [[Hyperbola#Hyperbola asAs an affine image of the hyperbola y = 1/x|hyperbolas]].

==See also==

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* {{citation|last=Young|first=John Wesley|title=Projective Geometry|year=1930|publisher=The Mathematical Association of America|series=The Carus Mathematical Monographs, Number Four}}

* {{citation | last1=van Yzeren | first1=Jan | title=A simple proof of Pascal's hexagon theorem |mr=1252929 | year=1993 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=100 | issue=10 | pages=930–931 | doi=10.2307/2324214 | publisher=Mathematical Association of America | jstor=2324214}}

{{Commons category|Pascal's hexagram}}

==External links==

{{Commons category|Pascal's hexagram}}

* [http://www.cut-the-knot.org/Curriculum/Geometry/Pascal.shtml Interactive demo of Pascal's theorem (Java required)] at [[cut-the-knot]]

* [http://www.cut-the-knot.org/Curriculum/Geometry/PascalLines.shtml 60 Pascal Lines (Java required)] at [[cut-the-knot]]

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* [http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes''] (PDF; 891&nbsp;kB), Uni Darmstadt, S. 29–35.

* [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.665.5892&rep=rep1&type=pdf How to Project Spherical Conics into the Plane] by Yoichi Maeda (Tokai University)

{{Blaise Pascal}}

{{Authority control}}

[[Category:Blaise Pascal]]