Pascal's theorem: Difference between revisions

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[[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 as an affine image of the hyperbola y=1/x|hyperbolas]].

The six vertices can be points at infinity themselves.

In the case of hyperbola:

* The secant line through a regular point and a point of infinity is the line passing the regular point and parallel to the corresponding asymptote.

* The secant line through both points of infinity is the line of infinity.

* The tangent line at a point of infinity is the corresponding asymptote.

In the case of parabola:

* The secant line through a regular point and the point of infinity is the line passing the regular point and parallel to axis.

* The tangent line at a point of infinity is the line of infinity.

==See also==