Interactive hyperbolic constructions: metric relations and billiards
DOI:
https://doi.org/10.35819/remat2022v8i2id5889Keywords:
Geometric Constructions, Hyperbolic Plane, Neutral Plane, Metric Relations, BilliardsAbstract
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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