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Submission declined on 14 April 2024 by ToadetteEdit (talk). This submission appears to be taken from https://groups.io/g/euclid/topic/96861800#5686. Wikipedia cannot accept material copied from elsewhere, unless it explicitly and verifiably has been released to the world under a suitably free and compatible copyright license or into the public domain and is written in an acceptable tone—this includes material that you own the copyright to. You should attribute the content of a draft to outside sources, using citations, but copying and pasting or closely paraphrasing sources is not acceptable. The entire draft should be written using your own words and structure. Declined by ToadetteEdit 7 months ago.This submission has now been cleaned of the above-noted copyright violation and its history redacted by an administrator to remove the infringement. If re-submitted (and subsequent additions do not reintroduce copyright problems), the content may be assessed on other grounds. |
The Aguilera-Brocard triangles are triangles of equal area that are formed with the Brocard points and three triangle centers on the Brocard axis. it derives its name from the work of Manuel Aguilera, a mathematics professor from Honduras, published in[1].
History
editIn 1881, French mathematician Henri Brocard publishes an article in the French Association for the Advancement of Science, giving rise to Brocard geometry by introducing the concepts of the Brocard circle and Brocard points[2]. At the beginning of the 20th century American Geometer Roger Arthur Johnson[3] showed that the Brocard points are symmetric with respect to the diameter of the Brocard circle. In the more recent past, all points created from a triangle are known as triangle centers following the publication of the Encyclopedia of Triangle Centers in the late 1990s by American Mathematician Clark Kimberling[4][5]. As history progressed, several points, apart from the circumcenter and Lemoine point, are found along the Brocard axis, which later became known as Kimberling centers in the Brocard axis[6]. The latter is crucial for the formation of the Aguilera-Brocard Triangles, as every Kimberling center on the Brocard axis aligns symmetrically with the Brocard points. These triangles can be expressed as:
The area of , Referring to the Brocard points as Ω1 and Ω2 we can designate two points and , on the Brocard axis with the aforementioned area. Let:
and , where and . The area of Ω1, and will be the area of Theorem 1.2 if
and . From the aforementioned equations for and , it can derive the following pairs of points { }, where each number in the square brackets represents a triangle center
Properties
editIn 2023, the Australian Mathematician Elias M Hagos[7] found an interesting result about the Aguilera-Brocard Triangles, as mentioned below:
Theorem 1. The Triangle Centers of the Aguilera-Brocard Triangles in the Brocard Axis are found in the Stothers Quintic (Q12)[8][9]
Finally, Greek Mathematician Antreas Hatzipolakis[10] discovered the following property:
Theorem 2. The isogonal conjugates of the triangle centers within the Aguilera-Brocard Triangles, situated along the Brocard axis, are found on the Kiepert circumhyperbola.
References
edit- ^ Aguilera, Manuel. "The Aguilera-Brocard Triangles". EUCLID. Encyclopedia of Triangle Centers. Retrieved 10 April 2024.
- ^ Brocard, H. (1881). Étude d'un nouveau cercle du plan du triangle. Association Française pour l'Académie des Sciences - Congrès d'Alger, 10, 138–159.
- ^ R. Johnson, Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle, Boston, MA: Houghton Mifflin, pp. 272-291, 1929.
- ^ Kimberling, C., & Lamoen, F. (1999). Central triangles. Nieuw Archief voor Wiskunde. Vierde Serie, 17.
- ^ Kimberling, C. (1998). Triangle centers and central triangles. Congr. Numer., 129, 1-295.
- ^ Weisstein, E. W. (2024). Brocard Axis. MathWorld--A Wolfram Web Resource. Retrieved [April 14, 2024], from https://mathworld.wolfram.com/BrocardAxis.html
- ^ Hagos M. Elias (2023). The Aguilera-Brocard Triangles. EUCLID message 5690. Encyclopedia of Triangle Centers. https://groups.io/g/euclid/message/5690 Retrieved 10 April 2024
- ^ Montesdeoca, A. (2022). Hechos Geometricos del Triangulo: Brocardianos y la Quintica Stothers. Recuperado de https://amontes.webs.ull.es/otrashtm/HGT2022.htm#HG101022
- ^ Gibert, B. (2024). Stothers Quintic Locus Properties. Retrieved from http://bernard-gibert.fr/curves/q012.html
- ^ Hatzipolakis, A. (2023, February 14). Re: Index Triangles in the ETC.
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