Abstract
Pappus' Involution Theorem is useful for proving incidence relations in the hyperbolic and elliptic planes. This fact is exemplified with the proof of a theorem about a family of 4-gons in the hyperbolic and elliptic planes. This non-Euclidean theorem is also re-interpreted in multiple ways, providing some other theorems for different figures in the hyperbolic plane.
Highlights
Involutions are a quite useful tool in theorem proving
A complete quadrangle is the figure in the projective plane that is produced by a set of four points, no three of which are collinear, when all the lines joining any two of them are drawn
Two sides of the complete quadrangle are opposite if they don’t share a vertex, and in this case they intersect in a diagonal point of the complete quadrangle
Summary
Involutions are a quite useful tool in theorem proving. Many geometric problems admit easier or shorter proofs using involutions (non-geometric problems too, —see [16] for an astonishing example!). During the author’s research [15], Pappus’ Involution Theorem was found as an extremely powerful tool for proving incidence relations in the hyperbolic and elliptic planes in the framework of Cayley–Klein models, and the first purpose of this article is to exhibit an example of this fact. Using Thurston’s trick, a unique projective proof of a non-Euclidean theorem about triangles provides a bunch of different nonEuclidean theorems, slightly different from the original one, which we call projective variations of the original one Another purpose of the paper is to illustrate this remarkable property of Cayley–Klein models, that seems to be not very well known, by exploring the projective variations of Theorem 1.2. In all figures right angles are denoted with the symbol
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