Constructions on the Poincaré-disk

Abstract

This paper concerns with implementations of constructions in the Poincaré model of hyperbolic geometry and their applications to proofs of selected theorems. The main motivation is how some hyperbolic geometric statements can be proven using elementary methods within the model. By embedding hyperbolic geometry into the Euclidean plane, certain proofs can become more accessible and comprehensible.

In this paper, we present two elementary constructions developed by using the Poincaré model, followed by novel-approached answers to the following questions. Does a common perpendicular always exist for two ultraparallel lines? Can a given line segment be divided into \(n\) equal parts? Is it possible to construct a triangle from three given angles, provided their sum is less than 180 degrees?

Although there exist known answers to these questions, the usual proofs involve strong theorems or trigonometric functions requiring extensive calculations (e.g. [6] and [2]). Instead, hereby we present proofs using elementary tools with easily understandable steps.