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dc.contributor.authorZięba, Maciej
dc.contributor.editorKrasiński, Tadeusz
dc.contributor.editorSpodzieja, Stanisław
dc.identifier.citationZięba 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.description.abstractIn 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.publisherWydawnictwo Uniwersytetu Łódzkiegopl_PL
dc.relation.ispartofAnalytic and Algebraic Geometry 3;
dc.rightsAttribution-NonCommercial-NoDerivatives 4.0 Międzynarodowe*
dc.titleA family of hyperbolas associated to a trianglepl_PL
dc.typeBook chapterpl_PL
dc.contributor.authorAffiliationPedagogical University of Cracow, Department of Mathematicspl_PL
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dc.referencesDecker, W.; Greuel, G.-M.; Pfister, G.; Schönemann, H.: Singular 4-1-2 — A computer algebra system for polynomial computations. (2019).pl_PL
dc.referencesMartel, S.: Eigenvectors, eigenvalues, and finite strain. Lecture notes University of Hawaii.pl_PL
dc.referencesMatthews, K. R.: Elementary linear algebra. Lecture Notes by Keith Matthews, Chapter 7: Identifying Second Degree Equations, pp. 129-148, (2013).pl_PL
dc.referencesWalter, Rolf: Lineare Algebra und Analytische Geometrie, Vieweg 1985pl_PL
dc.referencesWeisstein, E.: Steiner Inellipse. From MathWorld–A Wolfram Web Resource. http://
dc.referencesZięba, M.: O pewnych hiperbolach stowarzyszonych z trójkątem. In Polish. To appear in Prace Koła Matematyków Uniwersytetu Pedagogicznego w Krakowie.pl_PL
dc.referencesZięba, M.: Hyperbolas family. (Animation).

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Attribution-NonCommercial-NoDerivatives 4.0 Międzynarodowe
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