Poncelet's closure theorem

Illustration of Poncelet's porism for n = 3, a triangle that is inscribed in one circle and circumscribes another.

In geometry, Poncelet's closure theorem, also known as Poncelet's porism, states that whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics.[1][2] It is named after French engineer and mathematician Jean-Victor Poncelet, who wrote about it in 1822;[3] however, the triangular case was discovered significantly earlier, in 1746 by William Chapple.[4]

Poncelet's porism can be proved by an argument using an elliptic curve, whose points represent a combination of a line tangent to one conic and a crossing point of that line with the other conic.

  1. ^ Weisstein, Eric W. "Poncelet's Porism." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PonceletsPorism.html
  2. ^ King, Jonathan L. (1994). "Three problems in search of a measure". Amer. Math. Monthly. 101: 609–628. doi:10.2307/2974690.
  3. ^ Poncelet, Jean-Victor (1865) [1st. ed. 1822]. Traité des propriétés projectives des figures; ouvrage utile à ceux qui s'occupent des applications de la géométrie descriptive et d'opérations géométriques sur le terrain (in French) (2nd ed.). Paris: Gauthier-Villars. pp. 311–317.
  4. ^ Del Centina, Andrea (2016), "Poncelet's porism: a long story of renewed discoveries, I", Archive for History of Exact Sciences, 70 (1): 1–122, doi:10.1007/s00407-015-0163-y, MR 3437893