Construções hiperbólicas interativas: relações métricas e bilhares

Isabelle Siqueira da Costa; Marcel Vinhas Bertolini

doi:10.35819/remat2022v8i2id5889

Authors

Isabelle Siqueira da Costa Universidade Federal do Para (UFPA), Instituto de Ciências Exatas e Naturais, Belém, PA, Brasil https://orcid.org/0000-0002-7864-6384
Marcel Vinhas Bertolini Universidade Federal do Para (UFPA), Instituto de Ciências Exatas e Naturais, Belém, PA, Brasil https://orcid.org/0000-0002-1528-7660

DOI:

https://doi.org/10.35819/remat2022v8i2id5889

Keywords:

Geometric Constructions, Hyperbolic Plane, Neutral Plane, Metric Relations, Billiards

Abstract

This paper explores the geometry of hyperbolic and, more generaly, neutral planes, through straightedge and ruler constructions executed in the Poincaré disk in the software GeoGebra. The Theorems of Ceva and Euler are verifyed in the hyperbolic plane, besides metric relations associated to centroids and orthocenters. The usual technique of folding and unfolding billiards trajectories in polygonal regions is established in the neutral plane, motivated by the drawing of minimizing polygonal paths as, for example, in Fagnano's problem. This tool makes possible to describe billiards in stripes and, partially, in acutangle triangles, showing how its properties relate with the plane being euclidean or hyperbolic. An elementary proof is provided of an uniqueness property of the orthic trajectory in hyperbolic acutangle triangles, and complete proofs are given about orthic triangles in neutral planes.

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Author Biographies

Isabelle Siqueira da Costa, Universidade Federal do Para (UFPA), Instituto de Ciências Exatas e Naturais, Belém, PA, Brasil

http://lattes.cnpq.br/0048257402577122
Marcel Vinhas Bertolini, Universidade Federal do Para (UFPA), Instituto de Ciências Exatas e Naturais, Belém, PA, Brasil

http://lattes.cnpq.br/1495009615949354

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