Contents -- Preface -- 1. Euclidean Barycentric Coordinates and the Classic Triangle Centers -- 1.1 Points, Lines, Distance and Isometries -- 1.2 Vectors, Angles and Triangles -- 1.3 Euclidean Barycentric Coordinates -- 1.4 Analogies with Classical Mechanics -- 1.5 Barycentric Representations are Covariant -- 1.6 Vector Barycentric Representation -- 1.7 Triangle Centroid -- 1.8 Triangle Altitude -- 1.9 Triangle Orthocenter -- 1.10 Triangle Incenter -- 1.11 Triangle Inradius -- 1.12 Triangle Circumcenter -- 1.13 Circumradius -- 1.14 Triangle Incircle and Excircles -- 1.15 Excircle Tangency Points -- 1.16 From Triangle Tangency Points to Triangle Centers -- 1.17 Triangle In-Exradii -- 1.18 A Step Toward the Comparative Study -- 1.19 Tetrahedron Altitude -- 1.20 Tetrahedron Altitude Length -- 1.21 Exercises -- 2. Gyrovector Spaces and Cartesian Models of Hyperbolic Geometry -- 2.1 Einstein Addition -- 2.2 Einstein Gyration -- 2.3 From Einstein Velocity Addition to Gyrogroups -- 2.4 First Gyrogroup Theorems -- 2.5 The Two Basic Equations of Gyrogroups -- 2.6 Einstein Gyrovector Spaces -- 2.7 Gyrovector Spaces -- 2.8 Einstein Points, Gyrolines and Gyrodistance -- 2.9 Linking Einstein Addition to Hyperbolic Geometry -- 2.10 Einstein Gyrovectors, Gyroangles and Gyrotriangles -- 2.11 The Law of Gyrocosines -- 2.12 The SSS to AAA Conversion Law -- 2.13 Inequalities for Gyrotriangles -- 2.14 The AAA to SSS Conversion Law -- 2.15 The Law of Gyrosines -- 2.16 The ASA to SAS Conversion Law -- 2.17 Gyrotriangle Defect -- 2.18 Right Gyrotriangles -- 2.19 Einstein Gyrotrigonometry and Gyroarea -- 2.20 Gyrotriangle Gyroarea Addition Law -- 2.21 Gyrodistance Between a Point and a Gyroline -- 2.22 The Gyroangle Bisector Theorem -- 2.23 Mobius Addition and Mobius Gyrogroups -- 2.24 Mobius Gyration -- 2.25 Mobius Gyrovector Spaces.

2.26 Mobius Points, Gyrolines and Gyrodistance -- 2.27 Linking Mobius Addition to Hyperbolic Geometry -- 2.28 Mobius Gyrovectors, Gyroangles and Gyrotriangles -- 2.29 Gyrovector Space Isomorphism -- 2.30 Mobius Gyrotrigonometry -- 2.31 Exercises -- 3. The Interplay of Einstein Addition and Vector Addition -- 3.1 Extension of R into Tn+1 -- 3.2 Scalar Multiplication and Addition in Tn+1 -- 3.3 Inner Product and Norm in Tn+1 -- 3.4 Unit Elements of Tn+1 -- 3.5 From Tn+1 back to R -- 4. Hyperbolic Barycentric Coordinates and Hyperbolic Triangle Centers -- 4.1 Gyrobarycentric Coordinates in Einstein Gyrovector Spaces -- 4.2 Analogies with Relativistic Mechanics -- 4.3 Gyrobarycentric Coordinates in Mobius Gyrovector Spaces -- 4.4 Einstein Gyromidpoint -- 4.5 Mobius Gyromidpoint -- 4.6 Einstein Gyrotriangle Gyrocentroid -- 4.7 Einstein Gyrotetrahedron Gyrocentroid -- 4.8 Mobius Gyrotriangle Gyrocentroid -- 4.9 Mobius Gyrotetrahedron Gyrocentroid -- 4.10 Foot of a Gyrotriangle Gyroaltitude -- 4.11 Einstein Point to Gyroline Gyrodistance -- 4.12 Mobius Point to Gyroline Gyrodistance -- 4.13 Einstein Gyrotriangle Orthogyrocenter -- 4.14 Mobius Gyrotriangle Orthogyrocenter -- 4.15 Foot of a Gyrotriangle Gyroangle Bisector -- 4.16 Einstein Gyrotriangle Ingyrocenter -- 4.17 Ingyrocenter to Gyrotriangle Side Gyrodistance -- 4.18 Mobius Gyrotriangle Ingyrocenter -- 4.19 Einstein Gyrotriangle Circumgyrocenter -- 4.20 Einstein Gyrotriangle Circumgyroradius -- 4.21 Mobius Gyrotriangle Circumgyrocenter -- 4.22 Comparative Study of Gyrotriangle Gyrocenters -- 4.23 Exercises -- 5. Hyperbolic Incircles and Excircles -- 5.1 Einstein Gyrotriangle Ingyrocenter and Exgyrocenters -- 5.2 Einstein Ingyrocircle and Exgyrocircle Tangency Points -- 5.3 Useful Gyrotriangle Gyrotrigonometric Relations -- 5.4 The Tangency Points Expressed Gyrotrigonometrically.

5.5 M obius Gyrotriangle Ingyrocenter and Exgyrocenters -- 5.6 From Gyrotriangle Tangency Points to Gyrotriangle Gyrocenters -- 5.7 Exercises -- 6. Hyperbolic Tetrahedra -- 6.1 Gyrotetrahedron Gyroaltitude -- 6.2 Point Gyroplane Relations -- 6.3 Gyrotetrahedron Ingyrocenter and Exgyrocenters -- 6.4 In-Exgyrosphere Tangency Points -- 6.5 Gyrotrigonometric Gyrobarycentric Coordinates for the Gyrotetrahedron In-Exgyrocenters -- 6.6 Gyrotetrahedron Circumgyrocenter -- 6.7 Exercises -- 7. Comparative Patterns -- 7.1 Gyromidpoints and Gyrocentroids -- 7.2 Two and Three Dimensional Ingyrocenters -- 7.3 Two and Three Dimensional Circumgyrocenters -- 7.4 Tetrahedron Incenter and Excenters -- 7.5 Comparative study of the Pythagorean Theorem -- 7.6 Hyperbolic Heron's Formula -- 7.7 Exercises -- Notation And Special Symbols -- Bibliography -- Index.