TOPICS

# Pascal Lines

The lines containing the three points of the intersection of the three pairs of opposite sides of a (not necessarily regular) hexagon.

There are 6! (i.e., 6 factorial) possible ways of taking all polygon vertices in any order, but among these are six equivalent cyclic permutations and two possible orderings, so the total number of different hexagons (not all simple) is

There are therefore a total of 60 Pascal lines created by connecting polygon vertices in any order.

The 60 Pascal lines form a very complicated pattern which can be visualized most easily in the degenerate case of a regular hexagon inscribed in a circle, as illustrated above for magnifications ranging over five powers of 2. Only 45 lines are visible in this figure since each of the three thick lines (located at angles to each other) represents a degenerate group of four Pascal lines, and six of the Pascal lines are lines at infinity (Wells 1991).

The pattern for a general ellipse and hexagon (illustrated above) is much more complicated, and is difficult to distinguish from a clutter of lines.

The 60 Pascal lines intersect three at a time through 20 Steiner points (some of which are shown as the filled circles in the above figures). In the symmetrical case of the regular hexagon inscribed in a circle, the 20 Steiner points degenerate into seven distinct points arranged at the vertices and center of a regular hexagon centered at the origin of the circle. The 60 Pascal line also intersect three at a time in 60 Kirkman points. Each Steiner point lines together with three Kirkman points on a total of 20 Cayley lines. There is a dual relationship between the 60 Pascal lines and the 60 Kirkman points.

Cayley Lines, Heptagon Theorem, Hexagon, Kirkman Points, Pascal's Theorem, Plücker Lines, Salmon Points, Steiner Points

## Explore with Wolfram|Alpha

More things to try:

## References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 75, 1967.Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "The Heptagon Theorem." §2.1 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 8-11, 1974.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 236, 1929.Lachlan, R. "Pascal's Theorem." §181-191 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 113-119, 1893.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 172-173, 1991.

Pascal Lines

## Cite this as:

Weisstein, Eric W. "Pascal Lines." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PascalLines.html