Pascal's theorem: Difference between revisions

Revision as of 04:44, 27 January 2023

Revision as of 17:55, 31 October 2023

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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]], which 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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* {{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]]