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An introduction to geometry in the plane, both Euclidean and hyperbolic, this book is designed for an undergraduate course in geometry. With its patient approach, and plentiful illustrations, it will also be a stimulating read for anyone comfortable with the language of mathematical proof. While the material within is classical, it brings together topics that are not generally found together in books at this level, such as: parametric equations for the pseudosphere and its geodesics trilinear and barycentric coordinates Euclidean and hyperbolic tilings and theorems proved using inversion. The book is divided into four parts, and begins by developing neutral geometry in the spirit of Hilbert, leading to the Saccheri--Legendre Theorem. Subsequent sections explore classical Euclidean geometry, with an emphasis on concurrence results, followed by transformations in the Euclidean plane, and the geometry of the Poincaré disk model. -- Provided by publisher.
Axioms and models
Neutral Geometry
The axioms of incidence and order
Angles and triangles
Congruence verse I: SAS and ASA
Congruence verse II: AAS
Congruence verse III: SSS
Distance, length and the axioms of continuity
Angle measure
Triangles in neutral geometry
Polygons
Quadrilateral congruence theorems
Euclidean Geometry
The axiom on parallels
Parallel projection
Similarity
Circles
Circumference
Euclidean constructions
Concurrence I
Concurrence II
Concurrence III
Trilinear coordinates
Euclidean Transformations
Analytic geometry
Isometries
Reflections
Translations and rotations
Orientation
Glide reflections
Change of coordinates
Dilation
Applications of transformations
Area I
Area II
Barycentric coordinates
Inversion I
Inversion II
Applications of inversion
Hyperbolic Geometry
The search for a rectangle
Non-Euclidean parallels
The pseudosphere
Geodesics on the pseudosphere
The upper half-plane
The Poincaré disk
Hyperbolic reflections
Orientation preserving hyperbolic isometries
The six hyperbolic trigonometric functions
Hyperbolic trigonometry
Hyperbolic area
Tiling