dc.contributor.author | Zięba, Maciej | |
dc.contributor.editor | Krasiński, Tadeusz | |
dc.contributor.editor | Spodzieja, Stanisław | |
dc.date.accessioned | 2020-01-28T12:31:16Z | |
dc.date.available | 2020-01-28T12:31:16Z | |
dc.date.issued | 2019 | |
dc.identifier.citation | Zięba M., A family of hyperbolas associated to a triangle, in: Analytic and Algebraic Geometry 3, T. Krasiński, S. Spodzieja (red.), WUŁ, Łódź 2019, doi: 10.18778/8142-814-9.17. | pl_PL |
dc.identifier.isbn | 978-83-8142-814-9 | |
dc.identifier.uri | http://hdl.handle.net/11089/31348 | |
dc.description.abstract | In this note, we explore an apparently new one parameter family of conics associated to a triangle. Given a triangle we study ellipses whose one axis is parallel to one of sides of the triangle. The centers of these ellipses move along three hyperbolas, one for each side of the triangle. These hyperbolas intersect in four common points, which we identify as centers of incircle and the three excircles of the triangle. Thus they belong to a pencil of conics. We trace centers of all conics in the family and establish a surprising fact that they move along the excircle of the triangle. Even though our research is motivated by a problem in elementary geometry, its solution involves some non-trivial algebra and appeal to effective computational methods of algebraic geometry. Our work is illustrated by an animation in Geogebra and accompanied by a Singular file. | pl_PL |
dc.language.iso | en | pl_PL |
dc.publisher | Wydawnictwo Uniwersytetu Łódzkiego | pl_PL |
dc.relation.ispartof | Analytic and Algebraic Geometry 3; | |
dc.rights | Attribution-NonCommercial-NoDerivatives 4.0 Międzynarodowe | * |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | * |
dc.title | A family of hyperbolas associated to a triangle | pl_PL |
dc.type | Book chapter | pl_PL |
dc.page.number | 227-234 | pl_PL |
dc.contributor.authorAffiliation | Pedagogical University of Cracow, Department of Mathematics | pl_PL |
dc.identifier.eisbn | 978-83-8142-815-6 | |
dc.references | Ayoub, A. B.: The central conic sections revisited, Mathematics Magazine, 66 (5), (1993), 322 | pl_PL |
dc.references | Court, N. A.: Three Hyperbolas Associated with a Triangle, Amer. Math. Monthly, 64 (4), (1957), 241–247. | pl_PL |
dc.references | Decker, W.; Greuel, G.-M.; Pfister, G.; Schönemann, H.: Singular 4-1-2 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2019). | pl_PL |
dc.references | Martel, S.: Eigenvectors, eigenvalues, and finite strain. Lecture notes University of Hawaii. | pl_PL |
dc.references | Matthews, K. R.: Elementary linear algebra. Lecture Notes by Keith Matthews, Chapter 7: Identifying Second Degree Equations, pp. 129-148, (2013). | pl_PL |
dc.references | Walter, Rolf: Lineare Algebra und Analytische Geometrie, Vieweg 1985 | pl_PL |
dc.references | Weisstein, E.: Steiner Inellipse. From MathWorld–A Wolfram Web Resource. http:// mathworld.wolfram.com/SteinerInellipse.html | pl_PL |
dc.references | Zięba, M.: O pewnych hiperbolach stowarzyszonych z trójkątem. In Polish. To appear in Prace Koła Matematyków Uniwersytetu Pedagogicznego w Krakowie. | pl_PL |
dc.references | Zięba, M.: Hyperbolas family. (Animation). https://www.geogebra.org/m/f3muznrw | pl_PL |
dc.contributor.authorEmail | matematyka.maciej@gmail.com | pl_PL |
dc.identifier.doi | 10.18778/8142-814-9.17 | |