The 60 Pascal lines of a hexagon inscribed in a conic intersect three at a time through 20 Steiner points, and also three at a time in 60 points known as Kirkman points. Each Steiner point lies together with three Kirkman points on a total of 20 lines known as Cayley lines. There is a reciprocity relationship between the 60 Kirkman points and the 60 Pascal lines (Hesse, quoted in Salmon 1960), although the relationship is not one of duality in the commonly accepted meaning of that word.
Kirkman Points
See also
Cayley Lines, Pascal Lines, Pascal's Theorem, Plücker Lines, Salmon Points, Steiner PointsExplore with Wolfram|Alpha
References
Cremona, L. "Osservazioni sull'hexagrammum mysticum." Transunti della R. Acc. Nazionale dei Lincei 1, 142-143, 1876-77.Hesse, O. Vorlesungen über analytische Geometrie des Raumes. Leipzig, p. 186, 1861.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 236-237, 1929.Kirkman, T. P. Cambridge Dublin Math. J. 5, 185.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 116, 1893.Salmon, G. "Notes: Pascal's Theorem, Art. 267" in A Treatise on Conic Sections, 6th ed. New York: Chelsea, pp. 379-382, 1960.Veronese, G. "Nuovi teoremi sull'Hexagrammum mysticum." Transunti della R. Acc. Nazionale dei Lincei 1, 141-142, 1876-77.Veronese, G. "Nuovi teoremi sull'Hexagrammum mysticum." Mem. della R. Acc. Nazionale dei Lincei 1, 649-703, 1876-77.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 172, 1991.Referenced on Wolfram|Alpha
Kirkman PointsCite this as:
Weisstein, Eric W. "Kirkman Points." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KirkmanPoints.html