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<p> This paper concerns with implementations of constructions inthe Poincaré model of hyperbolic geometry and their applications to proofs ofselected theorems. The main motivation is how some hyperbolic geometric statements can be proven using elementary methods within the model. By embeddinghyperbolic geometry into the Euclidean plane, certain proofs can become moreaccessible and comprehensible.</p><p>In this paper, we present two elementary constructions developed by usingthe Poincaré model, followed by novel-approached answers to the following questions. Does a common perpendicular always exist for two ultraparallel lines? Cana given line segment be divided into <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1551_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation> equal parts? Is it possible to construct atriangle from three given angles, provided their sum is less than 180 degrees?</p><p>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 witheasily understandable steps.</p>

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Constructions on the Poincaré-disk

  • 期刊论文

  • 发表日期：

    2025年7月25日

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  • D. Veres

摘要

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

In this paper, we present two elementary constructions developed by usingthe Poincaré model, followed by novel-approached answers to the following questions. Does a common perpendicular always exist for two ultraparallel lines? Cana given line segment be divided into \(n\) n equal parts? Is it possible to construct atriangle 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 witheasily understandable steps.