Related papers: A hyperbolic proof of Pascal's Theorem

We provide hyperbolic analogues of some classical theorems in spherical geometry due to Menelaus, Euler, Lexell, Ceva and Lambert. Some of the spherical results are also made more precise.

We prove that a cyclic cover of a smooth complex projective variety is Brody hyperbolic if its branch divisor is a generic small deformation of a large enough multiple of a Brody hyperbolic base-point-free ample divisor. We also show the…

In this note, we present a proof of Smarandache's cevian triangle hyperbolic theorem in the Einstein relativistic velocity model of hyperbolic geometry.

This is a survey on Valiron's Theorem about the convergence properties of orbits of analytic self-maps of the disk of hyperbolic type and related questions in one and several variables.

We classify the polycyclic totally ordered simple dimension groups, i.e. dimension groups given by a dense embedding of n-dimensional lattice into the real line. Our method is based on the geometry of simple geodesics on the hyperbolic…

This papper aims to present and demonstrate Clifford's version for a generalization of Miquel's theorem with the use of Euclidean geometry arguments only.

We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry.

In this paper we give two different proofs of Bobenko and Springborn's theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface…

We discuss several ways of packing a hyperbolic surface with circles (of either varying radii or all being congruent) or horocycles, and note down some observations related to their symmetries (or the absence thereof).

A study of triangulations of cycles in the Cayley diagrams of finitely generated groups leads to a new geometric characterization of hyperbolic groups.

In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.

In this pedagogical note we present a short proof of the following main result of arxiv.org/abs/0911.5319, and clarify its relation to the isoperimetric problem. On the hyperbolic plane consider triangles ABC with fixed lengths of AB and…

The article deals with the connection between the second postulate of Euclid and non-Euclidean geometry. It is shown that the violation of the second postulate of Euclid inevitably leads to hyperbolic geometry. This eliminates…

There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a…

A soft presentation of hyperbolic spaces, free of differential apparatus, is offered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and…

Recently, Haase and Ilten initiated the study of classifying algebraically hyperbolic surfaces in toric threefolds. We complete this classification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, $\mathbb{P}^2 \times…

Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the…

We prove general results which include classical facts about 60 Pascal's lines as special cases. Along similar lines we establish analogous results about configurations of 2520 conics arising from Mystic Octagon. We offer a more…

We present a basic introduction to the theories of M\"obius structures and hyperbolic ends and we study their applications to the theory of $k$-surfaces in $3$-dimensional hyperbolic space.

Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to…