We study Poncelet type problem about the number of convex \(n\)-gons inscribed into one convex \(n\)-gon and circumscribed around another convex \(n\)-gon. It is proved that their number is at most 4. The result is generalized for spherical and hyperbolic geometries. This contrasts with Poncelet [4] type porisms where usually infinitude of such polygons is proved, provided that one such polygon already exists.

An inequality involving ratio of lengths of line segments is used. Alternative way of using Maclaurin–Braikenridge’s conic generation method [3], [1] generalized by Brianchon [2] is also discussed. Properties related to constructibility with straightedge and compass are also studied. A new proof, based on mathematical induction, of generalized Maclaurin–Braikenridge’s theorem is given. We also gave examples of regular polygons for which number 4 is realized.

References

[1] W. Braikenridge, Exercitatio Geometrica de Descriptione Curvarum, London, (1733).

[2] M. Brianchon, Solution de plusieurs problèmes de géométrie, Journal de l’École polytechnique, IX Cahier., Tome IV, 1-15, (1810).

[3] C. MacLaurin, A Letter from Mr. Colin Mac Laurin, Math. Prof. Edinburg. F.R.S. to Mr. John Machin, Ast. Prof. Gresh. & Secr. R.S. concerning the Description of Curve Lines’, Phil. Trans. 39, 143-165 (1735-36).

[4] J.V. Poncelet, Applications d’ analyse et de geometrie, Gauthier-Villars, in 2 vol. 1864 (1964).