Conic plane curve associated with a given triangle
In Euclidean geometry, a triangle conic is a conic in the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics. Other examples are the Steiner ellipse, which is an ellipse passing through the vertices and having its centre at the centroid of the reference triangle; the Kiepert hyperbola which is a conic passing through the vertices, the centroid and the orthocentre of the reference triangle; and the Artzt parabolas, which are parabolas touching two sidelines of the reference triangle at vertices of the triangle.
The terminology of triangle conic is widely used in the literature without a formal definition; that is, without precisely formulating the relations a conic should have with the reference triangle so as to qualify it to be called a triangle conic (see [1][2][3][4]). However, Greek mathematician Paris Pamfilos defines a triangle conic as a "conic circumscribing a triangle △ABC (that is, passing through its vertices) or inscribed in a triangle (that is, tangent to its side-lines)".[5][6] The terminology triangle circle (respectively, ellipse, hyperbola, parabola) is used to denote a circle (respectively, ellipse, hyperbola, parabola) associated with the reference triangle is some way.
Even though several triangle conics have been studied individually, there is no comprehensive encyclopedia or catalogue of triangle conics similar to Clark Kimberling's Encyclopedia of Triangle Centres or Bernard Gibert's Catalogue of Triangle Cubics.[7]
^Paris Pamfilos (2021). "Equilaterals Inscribed in Conics". International Journal of Geometry. 10 (1): 5–24.
^Christopher J Bradley. "Four Triangle Conics". Personal Home Pages. University of BATH. Retrieved 11 November 2021.
^Gotthard Weise (2012). "Generalization and Extension of the Wallace Theorem". Forum Geometricorum. 12: 1–11. Retrieved 12 November 2021.
^Zlatan Magajna. "OK Geometry Plus". OK Geometry Plus. Retrieved 12 November 2021.
^"Geometrikon". Paris Pamfilos home page on Geometry, Philosophy and Programming. Paris Palmfilos. Retrieved 11 November 2021.
^"1. Triangle conics". Paris Pamfilos home page on Geometry, Philosophy and Programming. Paris Palfilos. Retrieved 11 November 2021.
^Bernard Gibert. "Catalogue of Triangle Cubics". Cubics in Triangle Plane. Bernard Gibert. Retrieved 12 November 2021.
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