Pascal's Theorem

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Pascal's theorem

[pa′skalz ‚thir·əm]
(mathematics)
The theorem that when one inscribes a simple hexagon in a conic, the three pairs of opposite sides meet in collinear points.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Pascal’s Theorem

 

a theorem of geometry asserting that for any hexagon inscribed in a conic section (an ellipse, hyperbola, or parabola) the points of intersection of the three pairs of opposite sides (or of the sides’ extensions) lie on a single line called the Pascal line. The hexagon may be either convex or star-shaped. Figure 1 shows a hexagon whose vertices are successively numbered 1, 2, 3, 4, 5, and 6.

Figure 1

Sides that are separated from one another by two sides are considered opposite—that is, the sides 12 and 45, 23 and 56, and 34 and 61. Side 45, for example, is separated from side 12 by sides 23 and 34. The broken line is the Pascal line. If we number the vertices in a different order, that is, if we take a different hexagon, the Pascal line will also be different.

The theorem was established by B. Pascal in 1639. Pappus’ theorem, a special case of Pascal’s theorem for a pair of intersecting lines (a degenerate conic section), has been known since antiquity. This case is shown in Figure 2, where the vertices 1, 3, and 5 lie on one line and the vertices 2, 4, and 6 lie on the other

Figure 2

line; the broken line is the Pascal line. Pascal’s theorem is closely connected with Brianchon’s theorem. These theorems establish important projective properties of conic sections.

REFERENCES

Glagolev, N. A. Proektivnaia geometriia, 2nd ed. Moscow, 1963.
Efimov, N. V. Vysshaia geometriia, 5th ed. Moscow, 1971.
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
To this end we use the Pascal theorem ([12, Section 3.8]), which concerns a general hexagon inscribed in a circle.
The Pascal theorem holds for all kinds of inscribed hexagons including self-intersecting hexagon and this fact is used in the proof (figure 6).
According to the Pascal theorem the points E, Q and F lie on the same straight line.
The proof of the construction of the tangents was made by using tools of mathematics from different fields, and at its end--on the Pascal theorem. This situation indicates the importance of acquaintance with "forgotten" theorems in geometry.
According to the Pascal theorem, the points K, L and P lie on the same straight line.