The Hyperbolic Menelaus Theorem in The Poincaré Disc Model of Hyperbolic Geometry

In this note, we present the hyperbolic Menelaus theorem in the Poincaré disc of hyperbolic geometry. 2000 Mathematical Subject Classi…cation: 30F45, 20N99, 51B10, 51M10 Keywords and phrases: hyperbolic geometry, hyperbolic triangle, gyrovector 1. Introduction Hyperbolic Geometry appeared in the …rst half of the 19 century as an attempt to understand Euclid’s axiomatic basis of Geometry. It is also known as a type of nonEuclidean Geometry, being in many respects similar to Euclidean Geometry. Hyperbolic Geometry includes similar concepts as distance and angle. Both these geometries have many results in common but many are di¤erent. There are known many models for Hyperbolic Geometry, such as: Poincaré disc model, Poincaré half-plane, Klein model, Einstein relativistic velocity model, etc. The hyperbolic geometry is a non-euclidian geometry. Menelaus of Alexandria was a Greek mathematician and astronomer, the …rst to recognize geodesics on a curved surface as natural analogs of straight lines. Here, in this study, we present a proof of Menelaus’s theorem in the Poincaré disc model of hyperbolic geometry. The well-known Menelaus theorem states that if l is a line not through any vertex of a triangle ABC such that l meets BC in D; CA in E, and AB in F , then DB DC EC EA FA FB = 1 [1]. This result has a simple statement but it is of great interest. We just mention here few di¤erent proofs given by A. Johnson [2], N. A. Court [3], C. Coşni̧t1⁄4 a [4], A. Ungar [5]. We begin with the recall of some basic geometric notions and properties in the Poincaré disc. Let D denote the unit disc in the complex z plane, i.e. D = fz 2 C : jzj < 1g The most general Möbius transformation of D is z ! e z0 + z 1 + z0z = e (z0 z); which induces the Möbius addition in D, allowing the Möbius transformation of the disc to be viewed as a Möbius left gyro-translation z ! z0 z = z0 + z 1 + z0z followed by a rotation. Here 2 R is a real number, z; z0 2 D; and z0 is the complex conjugate of z0: Let Aut(D; ) be the automorphism group of the grupoid (D; ). If we de…ne gyr : D D ! Aut(D; ); gyr[a; b] = a b b a = 1 + ab 1 + ab ;