Reversion Porisms in Conics

Year 2021, Volume: 14 Issue: 2, 371 - 382, 29.10.2021

Lorenz Halbeısen , Norbert Hungerbühler , Marco Schiltknecht

https://doi.org/10.36890/iejg.979088

Abstract

We give a projective proof of the butterfly porism for cyclic quadrilaterals and present a general reversion porism for polygons with an arbitrary number of vertices on a conic. We also investigate projective properties of the porisms.

Keywords

Porisms, butterfly theorems, reversion map

References

• [1] Bogomolny, A.: The butterfly theorem. Interactive Mathematics Miscellany and Puzzles. http://www.cut-the-knot.org/pythagoras/Butterfly.shtml, accessed February 4, 2021.
• [2] Coxeter, H. S. M., Greitzer, S. L.: Geometry revisited, volume 19 of New Mathematical Library. Random House, Inc. New York (1967).
• [3] Izmestiev, I.: A porism for cyclic quadrilaterals, butterfly theorems, and hyperbolic geometry. Amer. Math. Monthly. 122 (5), 467–475 (2015).
• [4] Jones, D.: Quadrangles, butterflies, Pascal’s hexagon, and projective fixed points. Amer. Math. Monthly. 87 (3), 197–200 (1980).
• [5] Klamkin, M. S.: An Extension of the Butterfly Problem. Math. Mag. 38 (4), 206–208 (1965).
• [6] Kocik, J.: A porism concerning cyclic quadrilaterals. Geometry, Article ID 483727: 5 pages (2013).
• [7] Sliepčević, A.: A new generalization of the butterfly theorem. J. Geom. Graph. 6 (1), 61–68 (2002).
• [8] Volenec, V.: A generalization of the butterfly theorem. Math. Commun. 5 (2), 157–160 (2000).