Menelauss Theorem for Hyperbolic Quadrilaterals in The Einstein Relativistic Velocity Model of Hyperbolic Geometry

In this study, we present (i) a proof of the Menelaus theorem for quadrilaterals in hyperbolic geometry, (ii) and a proof for the transversal theorem for triangles, and (iii) the Menelaus’s theorem for n-gons 2000 Mathematical Subject Classi…cation: 51K05, 51M10, 30F45, 20N99, 51B10 Keywords and phrases: hyperbolic geometry, hyperbolic triangle, hyperbolic quadrilateral, Menelaus theorem, transversal theorem, gyrovector 1 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 non-Euclidean 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: Poincare disc model, Poincare halfplane, Klein model, Einstein relativistic velocity model, etc. 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 give hyperbolic version of Menelaus theorem for quadrilaterals. 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]. F. Smarandache (1983) has generalized the Theorem of Menelaus for any polygon with n 4 sides as follows: If a line l intersects the n-gon A1A2:::An sides A1A2; A2A3; :::; and AnA1 respectively in the points M1;M2; :::; and Mn, then M1A1 M1A2 M2A2 M2A3 ::: MnAn MnA1 = 1 [2]. Let D denote the complex unit disc in complex z plane, i.e. D = fz 2 C : jzj < 1g: The most general Mobius transformation of D is z ! e z0 + z 1 + z0z = e (z0 z);