Draft:Aguilera–Brocard triangle

The Aguilera-Brocard Triangles in the Brocard Circle

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

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In 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

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In 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

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  1. ^ Aguilera, Manuel. "The Aguilera-Brocard Triangles". EUCLID. Encyclopedia of Triangle Centers. Retrieved 10 April 2024.
  2. ^ 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.
  3. ^ R. Johnson, Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle, Boston, MA: Houghton Mifflin, pp. 272-291, 1929.
  4. ^ Kimberling, C., & Lamoen, F. (1999). Central triangles. Nieuw Archief voor Wiskunde. Vierde Serie, 17.
  5. ^ Kimberling, C. (1998). Triangle centers and central triangles. Congr. Numer., 129, 1-295.
  6. ^ Weisstein, E. W. (2024). Brocard Axis. MathWorld--A Wolfram Web Resource. Retrieved [April 14, 2024], from https://mathworld.wolfram.com/BrocardAxis.html
  7. ^ 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
  8. ^ Montesdeoca, A. (2022). Hechos Geometricos del Triangulo: Brocardianos y la Quintica Stothers. Recuperado de https://amontes.webs.ull.es/otrashtm/HGT2022.htm#HG101022
  9. ^ Gibert, B. (2024). Stothers Quintic Locus Properties. Retrieved from http://bernard-gibert.fr/curves/q012.html
  10. ^ Hatzipolakis, A. (2023, February 14). Re: Index Triangles in the ETC.