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Preface | |

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Author | |

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Lines, Distance, Segments, and Rays | |

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Intended Goals | |

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Axioms of Alignment | |

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A Glimpse at Finite Geometry | |

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Metric Geometry | |

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Eves' 25-Point Affine Geometry: A Model for Axioms 0-4 | |

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Distance and Alignment | |

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Properties of Betweenness: Segments and Rays | |

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Coordinates for Rays | |

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Geometry and the Continuum | |

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Segment Construction Theorems | |

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Angles, Angle Measure, and Plane Separation | |

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Angles and Angle Measure | |

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Plane Separation | |

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Consequences of Plane Separation: The Postulate of Pasch | |

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The Interior of an Angle: The Angle Addition Postulate | |

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Angle Construction Theorems | |

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Consequences of a Finite Metric | |

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Unified Geometry: Triangles and Congruence | |

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Congruent Triangles: SAS Hypothesis | |

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A Metric for City Centers | |

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The SAS Postulate and the ASA and SSS Theorems | |

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Euclid's Superposition Proof: An Alternative to Axiom 12 | |

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Locus, Perpendicular Bisectors, and Symmetry | |

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The Exterior Angle Inequality | |

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Inequalities for Triangles | |

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Further Congruence Criteria | |

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Special Segments Associated with Triangles | |

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Quadrilaterals, Polygons, and Circles | |

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Quadrilaterals | |

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Congruence Theorems for Convex Quadrilaterals | |

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The Quadrilaterals of Saccheri and Lambert | |

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Polygons | |

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Circles in Unified Geometry | |

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Three Geometries | |

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Parallelism in Unified Geometry and the Influence of ï¿½ | |

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Elliptic Geometry: Angle-Sum Theorem | |

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Pole-Polar Theory for Elliptic Geometry | |

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Angle Measure and Distance Related: Archimedes' Method | |

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Hyperbolic Geometry: Angle-Sum Theorem | |

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A Concept for Area: AAA Congruence | |

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Parallelism in Hyperbolic Geometry | |

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Asymptotic Triangles in Hyperbolic Geometry | |

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Euclidean Geometry: Angle-Sum Theorem | |

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Median of a Trapezoid in Euclidean Geometry | |

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Similar Triangles in Euclidean Geometry | |

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Pythagorean Theorem | |

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Inequalities for Quadrilaterals: Unified Trigonometry | |

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An Inequality Concept for Unified Geometry | |

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Ratio Inequalities for Trapezoids | |

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Ratio Inequalities for Right Triangles | |

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Orthogonal Projection and ï¿½Similarï¿½ Triangles | |

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Unified Trigonometry: The Functions c(ï¿½) and s(ï¿½) | |

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Trigonometric identities | |

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Classical Forms for c(ï¿½) and s(ï¿½) | |

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Lambert Quadrilaterals and the Function C(u) | |

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Identities for C(u) | |

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Classical Forms for C(u) | |

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The Pythagorean Relation for Unified Geometry | |

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Classical Unified Trigonometry | |

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Beyond Euclid: Modern Geometry | |

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Directed Distance: Stewart's Theorem and the Cevian Formula | |

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Formulas for Special Cevians | |

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Circles: Power Theorems and Inscribed Angles | |

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Using Circles in Geometry | |

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Cross Ratio and Harmonic Conjugates | |

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The Theorems of Ceva and Menelaus | |

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Families of Mutually Orthogonal Circles | |

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Transformations in Modern Geometry | |

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Projective Transformations | |

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Affine Transformations | |

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Similitudes and Isometries | |

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Line Reflections: Building Blocks for Isometries and Similitudes | |

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Translations and Rotations | |

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Circular Inversion | |

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Non-Euclidean Geometry: Analytical Approach | |

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Law of Sines and Cosines for Unified Geometry | |

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Unifying Identities for Unified Trigonometry | |

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Half-Angle Identities for Unified Geometry | |

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The Shape of a Triangle in Unified Geometry: Cosine Inequality | |

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The Equations of Gauss: Area of a Triangle | |

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Directed Distance: Theorems of Menelaus and Ceva | |

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Poincarï¿½'s Model for Hyperbolic Geometry | |

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Other Models: Surface Theory | |

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Hyperbolic Parallelism and Bolyai's Ideal Points | |

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Sketchpad Experiments | |

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Intuitive Spherical Geometry | |

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Proof in Geometry | |

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The Real Numbers and Least Upper Bound | |

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Floating Triangles/Quadrilaterals | |

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Axiom Systems for Geometry | |

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Solutions to Selected Problems | |

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Bibliography | |

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Index | |