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Keywords = brocard points

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15 pages, 476 KB  
Article
A Special Family of Triangles
by Ivana Božić Dragun, Helena Koncul and Boris Odehnal
Mathematics 2025, 13(17), 2808; https://doi.org/10.3390/math13172808 - 1 Sep 2025
Viewed by 1365
Abstract
We study the one-parameter family of triangles that emerges if one sideline traces a pencil of lines and the opposite angle is fixed. A description of the traces of triangle centers and the pair of Brocard points in terms of parametrizations and equations [...] Read more.
We study the one-parameter family of triangles that emerges if one sideline traces a pencil of lines and the opposite angle is fixed. A description of the traces of triangle centers and the pair of Brocard points in terms of parametrizations and equations is given. The envelopes of the families of circumcircles and nine-point circles are determined. Our approach even allows us to consider and treat the triangle family as a two-parameter family of triangles; i.e., the (interior) angle opposite to the pencil, which is in the beginning fixed, may also change. Full article
(This article belongs to the Section B: Geometry and Topology)
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10 pages, 2599 KB  
Article
Poncelet Porisms and Loci of Centers in the Isotropic Plane
by Ema Jurkin
Mathematics 2024, 12(4), 618; https://doi.org/10.3390/math12040618 - 19 Feb 2024
Cited by 2 | Viewed by 2069
Abstract
Any triangle in an isotropic plane has a circumcircle u and incircle i. It turns out that there are infinitely many triangles with the same circumcircle u and incircle i. This one-parameter family of triangles is called a poristic system of [...] Read more.
Any triangle in an isotropic plane has a circumcircle u and incircle i. It turns out that there are infinitely many triangles with the same circumcircle u and incircle i. This one-parameter family of triangles is called a poristic system of triangles. We study the trace of the centroid, the Feuerbach point, the symmedian point, the Gergonne point, the Steiner point and the Brocard points for such a system of triangles. We also study the traces of some further points associated with the triangles of the poristic family, and we prove that the vertices of the contact triangle, tangential triangle and anticomplementary triangle move on circles while the initial triangle traverses the poristic family. Full article
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13 pages, 323 KB  
Article
On Some Properties of the First Brocard Triangle in the Isotropic Plane
by Vladimir Volenec, Zdenka Kolar-Begović and Ružica Kolar-Šuper
Mathematics 2022, 10(9), 1381; https://doi.org/10.3390/math10091381 - 20 Apr 2022
Viewed by 3081
Abstract
In this paper we introduce the first Brocard triangle of an allowable triangle in the isotropic plane and derive the coordinates of its vertices in the case of a standard triangle. We prove that the first Brocard triangle is homologous to the given [...] Read more.
In this paper we introduce the first Brocard triangle of an allowable triangle in the isotropic plane and derive the coordinates of its vertices in the case of a standard triangle. We prove that the first Brocard triangle is homologous to the given triangle and that these two triangles are parallelogic. We consider the relationships between the first Brocard triangle and the Steiner axis, the Steiner point, and the Kiepert parabola of the triangle. We also investigate some other interesting properties of this triangle and consider relationships between the Euclidean and the isotropic case. Full article
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