Pascal's theorem

thumb|250px|Pascal line of self-crossing hexagon inscribed in ellipse. Opposite sides of hexagon have the same color.

thumb|250px|Self-crossing hexagon , 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 (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 sides of the hexagon (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, which is the special case of a degenerate conic of two lines with three points on each line.

Euclidean variants

The most natural setting for Pascal's theorem is in a projective plane since any two lines meet and no exceptions need to be made for parallel lines. However, the theorem remains valid in the Euclidean plane, with the correct interpretation of what happens when some opposite sides of the hexagon are parallel.

If exactly one pair of opposite sides of the hexagon is parallel, then the conclusion of the theorem is that the "Pascal line" determined by the two points of intersection is parallel to the parallel sides of the hexagon. If two pairs of opposite sides are parallel, then the third pair of opposite sides is also parallel, and there is no Pascal line in the Euclidean plane (in this case, the line at infinity of the extended Euclidean plane is the Pascal line of the hexagon).

Pascal's theorem is the 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 titled "Essay pour les coniques. Par B. P."

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 on a conic , the intersection of alternate sides, , , together with the intersection of tangents at opposite vertices and 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-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 , 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. The Braikenridge–Maclaurin theorem may be applied in the Braikenridge–Maclaurin construction, which is a 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 sides is inscribed in a conic section, and opposite pairs of sides are extended until they meet in points. Then if of those points lie on a common line, the last point will be on that line, too.

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 configuration of 60 lines is called the Hexagrammum Mysticum.

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. 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.

Proofs

Pascal's original note

:<math>\frac{\overline{GB{\overline{GA \times \frac{\overline{HA{\overline{HF \times \frac{\overline{KF{\overline{KE \times\frac{\overline{GE{\overline{GD \times \frac{\overline{HD{\overline{HC \times \frac{\overline{KC{\overline{KB=1.</math>

Degenerations of Pascal's theorem

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 parabolas and hyperbolas.

Notes

References

External links

Interactive demo of Pascal's theorem (Java required) at cut-the-knot
60 Pascal Lines (Java required) at cut-the-knot
The Complete Pascal Figure Graphically Presented by J. Chris Fisher and Norma Fuller (University of Regina)
Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes (PDF; 891 kB), Uni Darmstadt, S. 29–35.
How to Project Spherical Conics into the Plane by Yoichi Maeda (Tokai University)