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| Introduction | |
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| The Geometry of Our World | |
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| A Review of Terminology | |
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| Notes on Notation | |
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| Notes on the Exercises | |
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| Euclidean Geometry | |
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| The Pythagorean Theorem | |
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| The Axioms of Euclidean Geometry | |
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| SSS, SAS, and ASA | |
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| Parallel Lines | |
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| Pons Asinorum | |
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| The Star Trek Lemma | |
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| Similar Triangles | |
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| Power of the Point | |
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| The Medians and Centroid | |
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| The Incircle, Excircles, and the Law of Cosines | |
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| The Circumcircle and the Law of Sines | |
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| The Euler Line | |
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| The Nine Point Circle | |
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| Pedal Triangles and the Simson Line | |
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| Menelaus and Ceva | |
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| Geometry in Greek Astronomy | |
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| The Relative Size of the Moon and Sun | |
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| The Diameter of the Earth | |
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| The Babylonians to Kepler, a Time Line | |
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| Constructions Using a Compass and Straightedge | |
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| The Rules | |
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| Some Examples | |
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| Basic Results | |
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| The Algebra of Constructible Lengths | |
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| The Regular Pentagon | |
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| Other Constructible Figures | |
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| Trisecting an Arbitrary Angle | |
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| Geometer's Sketchpad | |
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| The Rules of Constructions | |
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| Lemmas and Theorems | |
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| Archimedes' Trisection Algorithm | |
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| Verification of Theorems | |
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| Sophisticated Results | |
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| Parabola Paper | |
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| Higher Dimensional Objects | |
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| The Platonic Solids | |
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| The Duality of Platonic Solids | |
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| The Euler Characteristic | |
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| Semiregular Polyhedra | |
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| A Partial Categorization of Semiregular Polyhedra | |
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| Four-Dimensional Objects | |
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| Hyperbolic Geometry | |
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| Models | |
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| Results from Neutral Geometry | |
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| The Congruence of Similar Triangles | |
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| Parallel and Ultraparallel Lines | |
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| Singly Asymptotic Triangles | |
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| Doubly and Triply Asymptotic Triangles | |
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| The Area of Asymptotic Triangles | |
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| The Poincare Models of Hyperbolic Geometry | |
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| The Poincare Upper Half Plane Model | |
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| Vertical (Euclidean) Lines | |
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| Isometries | |
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| Inversion in the Circle | |
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| Inversion in Euclidean Geometry | |
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| Fractional Linear Transformations | |
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| The Cross Ratio | |
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| Translations | |
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| Rotations | |
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| Reflections | |
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| Lengths | |
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| The Axioms of Hyperbolic Geometry | |
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| The Area of Triangles | |
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| The Poincare Disc Model | |
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| Circles and Horocycles | |
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| Hyperbolic Trigonometry | |
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| The Angle of Parallelism | |
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| Curvature | |
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| Tilings and Lattices | |
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| Regular Tilings | |
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| Semiregular Tilings | |
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| Lattices and Fundamental Domains | |
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| Tilings in Hyperbolic Space | |
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| Tilings in Art | |
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| Foundations | |
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| Theories | |
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| The Real Line | |
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| The Plane | |
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| Line Segments and Lines | |
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| Separation Axioms | |
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| Circles | |
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| Isometries and Congruence | |
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| The Parallel Postulate | |
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| Similar Triangles | |
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| Spherical Geometry | |
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| The Area of Triangles | |
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| The Geometry of Right Triangles | |
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| The Geometry of Spherical Triangles | |
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| Menelaus' Theorem | |
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| Heron's Formula | |
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| Tilings of the Sphere | |
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| The Axioms | |
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| Elliptic Geometry | |
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| Projective Geometry | |
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| Moving a Line to Infinity | |
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| Pascal's Theorem | |
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| Projective Coordinates | |
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| Duality | |
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| Dual Conics and Brianchon's Theorem | |
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| Areal Coordinates | |
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| The Pseudosphere in Lorentz Space | |
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| The Sphere as a Foil | |
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| The Pseudosphere | |
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| Angles and the Lorentz Cross Product | |
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| A Different Perspective | |
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| The Beltrami-Klein Model | |
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| Menelaus' Theorem | |
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| Finite Geometry | |
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| Algebraic Affine Planes | |
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| Algebraic Projective Planes | |
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| Weak Incidence Geometry | |
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| Geometric Projective Planes | |
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| Addition | |
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| Multiplication | |
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| The Distributive Law | |
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| Commutativity, Coordinates, and Pappus' Theorem | |
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| Weak Projective Space and Desargues' Theorem | |
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| Nonconstructibility | |
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| The Field of Constructible Numbers | |
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| Fields as Vector Spaces | |
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| The Field of Definition for a Construction | |
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| The Regular 7-gon | |
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| The Regular 17-gon | |
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| Modern Research in Geometry | |
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| Pythagorean Triples | |
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| Bezout's Theorem | |
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| Elliptic Curves | |
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| A Mixture of Cevians | |
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| A Challenge for Fermat | |
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| The Euler Characteristic in Algebraic Geometry | |
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| Lattice Point Problems | |
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| Fractals and the Apollonian Packing Problem. S | |