Polynomial triangle centers on the line at infinity

Points on the line at infinity in the extended plane of a triangle ABC are discussed in terms of barycentric coordinates that are polynomials in the sidelengths a ,  b ,  c . Various properties of the line at infinity are discussed, including two theorems, with related conjectures, on polynomial rep...

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Bibliographic Details
Published inJournal of geometry Vol. 111; no. 1
Main Author Kimberling, Clark
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.04.2020
Springer Nature B.V
Subjects
Geometry
Infinity
Mathematics
Mathematics and Statistics
Polynomials
isogonal conjugate
Primary 51N20
line at infinity
Nagel line
Secondary 51N15
barycentric coordinates
isotomic conjugate
circumcircle
symbolic substitution
Triangle center
polynomial triangle center
Euler line
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Summary:Points on the line at infinity in the extended plane of a triangle ABC are discussed in terms of barycentric coordinates that are polynomials in the sidelengths a ,  b ,  c . Various properties of the line at infinity are discussed, including two theorems, with related conjectures, on polynomial representations of triangle centers that are at opposite ends of a diameter of the circumcircle—along with their isogonal conjugates on the line at infinity. Also considered are an equal-areas locus, symbolic substitution, and historical comments.
ISSN:0047-2468
1420-8997
DOI:10.1007/s00022-020-0522-y