# Difference between revisions of "Pascal theorem"

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The opposite sides of a hexagon (cf. [[Polygon|Polygon]]) inscribed in a second-order curve (a [[Conic|conic]]) intersect in three collinear points (on the Pascal line, see Fig. a). When two adjacent vertices coincide, the line passing through them is understood to be the tangent to the conic at that point. The tangent to a conic drawn at one of the vertices of an inscribed pentagon intersects the side opposite to this vertex at a point on the line passing through the points of intersection of the remaining pairs of non-adjacent sides of the pentagon (see Fig. b). | The opposite sides of a hexagon (cf. [[Polygon|Polygon]]) inscribed in a second-order curve (a [[Conic|conic]]) intersect in three collinear points (on the Pascal line, see Fig. a). When two adjacent vertices coincide, the line passing through them is understood to be the tangent to the conic at that point. The tangent to a conic drawn at one of the vertices of an inscribed pentagon intersects the side opposite to this vertex at a point on the line passing through the points of intersection of the remaining pairs of non-adjacent sides of the pentagon (see Fig. b). | ||

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Figure: p071780b | Figure: p071780b | ||

− | If | + | If $ABCD$ is a quadrangle inscribed in a conic, then the points of intersection of the tangents at $C$ and $D$ with the sides $AD$ and $BC$, respectively, and the point of intersection of $AB$ and $CD$ are collinear (see Fig. c). |

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071780c.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p071780c.gif" /> |

## Latest revision as of 23:10, 11 April 2014

The opposite sides of a hexagon (cf. Polygon) inscribed in a second-order curve (a conic) intersect in three collinear points (on the Pascal line, see Fig. a). When two adjacent vertices coincide, the line passing through them is understood to be the tangent to the conic at that point. The tangent to a conic drawn at one of the vertices of an inscribed pentagon intersects the side opposite to this vertex at a point on the line passing through the points of intersection of the remaining pairs of non-adjacent sides of the pentagon (see Fig. b).

Figure: p071780a

Figure: p071780b

If $ABCD$ is a quadrangle inscribed in a conic, then the points of intersection of the tangents at $C$ and $D$ with the sides $AD$ and $BC$, respectively, and the point of intersection of $AB$ and $CD$ are collinear (see Fig. c).

Figure: p071780c

The points of intersection of the tangent at the vertices of a triangle inscribed in a conic with the opposite sides are collinear (see Fig. d).

Figure: p071780d

The dual to Pascal's theorem is the Brianchon theorem. B. Pascal established his theorem in 1639. The special case of a conic degenerating to a pair of lines was known even in Antiquity (see Pappus axiom).

#### References

[1] | N.A. Glagolev, "Projective geometry" , Moscow (1963) (In Russian) |

[2] | N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian) |

#### Comments

#### References

[a1] | D. Hilbert, "Grundlagen der Geometrie" , Teubner, reprint (1968) |

[a2] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) pp. 185–186; 396–404 |

[a3] | H.S.M. Coxeter, "Projective geometry" , Univ. Toronto Press (1987) pp. 85–90; 145–146 |

[a4] | G. Salmon, "A treatise on conic sections" , Longman (1879) pp. 267 |

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Pascal theorem.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Pascal_theorem&oldid=14328