Smarandaches Cevian Triangle Theorem in The Einstein Reletivistic Velocity Model of Hyperbolic Geometry

In this note, we present a proof of Smarandache’s cevian triangle hyperbolic theorem in the Einstein relativistic velocity model of hyperbolic geometry. 2000 Mathematical Subject Classi…cation: 51K05, 51M10, 30F45, 20N99, 51B10 Keywords and phrases: hyperbolic geometry, hyperbolic triangle, Smarandache’s cevian triangle, gyrovector, Einstein relativistic velocity model 1. Introduction Hyperbolic geometry appeared in the …rst half of the 19 century as an attempt to understand Euclid’s axiomatic basis for geometry. It is also known as a type of nonEuclidean geometry, being in many respects similar to Euclidean geometry. Hyperbolic geometry includes such concepts as: distance, angle and both of them have many theorems in common.There are known many main models for hyperbolic geometry, such as: Poincare disc model, Poincare half-plane, Klein model, Einstein relativistic velocity model, etc. The hyperbolic geometry is a non-Euclidian geometry. Here, in this study, we present a proof of Smarandache’s cevian triangle hyperbolic theorem in the Einstein relativistic velocity model of hyperbolic geometry. Smarandache’s cevian triangle theorem states that if A1B1C1 is the cevian triangle of point P with respect to the triangle ABC; then PA PA1 PB PB1 PC PC1 = AB BC CA A1B B1C C1A [1]. 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); which induces the Mobius addition in D, allowing the Mobius transformation of the disc to be viewed as a Mobius left gyrotranslation 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 ; then is true gyrocommutative law a b = gyr[a; b](b a):