Triangles with arithmetic, geometric and harmonic progressions

Abstract

This article explores the fascinating world of triangles with angle and side sequences in arithmetic, geometric and harmonic progressions. The properties and relationships between these progressions and the corresponding triangle parameters are investigated. The main focus is on proving a series of theorems that provide insights into the conditions under which triangles exhibit these progressions. The proofs are presented with meticulous attention to detail, offering a deep understanding of the results. The presented findings contribute to the field of mathematical education by providing rich material for classroom activities and discussions.

Keywords:

• Arithmetic progressions
• geometric progressions
• harmonic progressions
• triangles
• geometric properties
• trigonometric relationships
• mathematical education

Acknowledgments

We would like to express our sincere appreciation to the anonymous reviewer for their thoughtful comments and valuable insights, which significantly contributed to the improvement of this paper.

This work is the outcome of a Master's dissertation conducted within the framework of PROFMAT (Programa de Mestrado Profissional de Matemática em Rede), a program that plays a pivotal role in the professional development of Mathematics teachers in Brazil. We extend our heartfelt gratitude to the faculty and administrators of PROFMAT for their support and guidance throughout this academic journey.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

We also wish to express our thanks to CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) for their financial support, which made this research possible.