The Barycenter as the Critical Point of the Geometric Mean function between two specific distances in any triangle

Authors

DOI:

https://doi.org/10.22481/intermaths.v5i1.14238

Keywords:

Median, Centroid, Optmization, Geometric Mean

Abstract

The present article aims to study the centroid of a triangle as a critical point of a function. The considered function is the geometric mean between two specific distances within the triangle. In the demonstration, the Law of Cosines, Stewart's Theorem, and Differential Calculus are employed. The problem is verified using the GeoGebra software. It has the potential to be used in Geometry or Differential Calculus classes in Higher Education.

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Author Biographies

Rogério César Santos, Universidade de Brasília, Brasília, DF, Brasil.

Licenciado em Matemática pela Universidade de Brasília em 2001. Mestre em Matemática pela Universidade de Brasília em 2003. Doutor em Educação pela Universidade de Brasília em 2018. Docente da Universidade de Brasília desde 2010.

Ruthyelen Cristina Machado de Freitas, Secretaria de Estado de Educação do Distrito Federal, Brasília, DF, Brasil.

Possui Graduação em Matemática pela Universidade Estadual de Goiás (2010). É Professora da Secretaria de Educação do Distrito Federal. É discente do Programa de Pós-Graduação do Mestrado Profissional em Matemática em Rede Nacional - Profmat do Distrito Federal.

References

Bialostocki, A. and Bialostocki, D. ``The incenter and an excenter as solutions to an extremal problem", textit{Forum Geometricorum}, 11, 9–12, 2011.

Hajja, M. ``Extremal properties of the incentre and the excentres of a triangle", textit{The Mathematical Gazette}, 2012, Vol. 96, No. 536, pp. 315-317, 2012.

Bialostocki, A. and Ely, R. ``Points on a line that maximize and minimize the ratio of the distances to two given lines", textit{Forum Geometricorum}, 15, 177-178, 2015.

Hajja, M. ``One more note on the extremal properties of the incentre and the excentres of a triangle", textit{The Mathematical Gazette}, Vol. 101, No. 551, pp. 308-310, 2017.

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Published

2024-06-30

How to Cite

Santos, R. C., & de Freitas, R. C. M. (2024). The Barycenter as the Critical Point of the Geometric Mean function between two specific distances in any triangle. INTERMATHS, 5(1), 108-117. https://doi.org/10.22481/intermaths.v5i1.14238

Issue

Vol. 5 No. 1 (2024)

Section

Artigos

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