Lines, Distance, Segments, and Rays Intended Goals Axioms of Alignment A Glimpse at Finite Geometry Metric Geometry Eves'' 25-Point Affine Geometry: A Model for Axioms 0-4 Distance and Alignment Properties of Betweenness: Segments and Rays Coordinates for Rays Geometry and the Continuum Segment Construction Theorems Angles, Angle Measure, and Plane Separation Angles and Angle Measure Plane Separation Consequences of Plane Separation: The Postulate of Pasch The Interior of an Angle: The Angle Addition Postulate Angle Construction Theorems Consequences of a Finite Metric Unified Geometry: Triangles and Congruence Congruent Triangles: SAS Hypothesis A Metric for City Centers The SAS Postulate and the ASA and SSS Theorems Euclid''s Superposition Proof: An Alternative to Axiom 12 Locus, Perpendicular Bisectors, and Symmetry The Exterior Angle Inequality Inequalities for Triangles Further Congruence Criteria Special Segments Associated with Triangles Quadrilaterals, Polygons, and Circles Quadrilaterals Congruence Theorems for Convex Quadrilaterals The Quadrilaterals of Saccheri and Lambert Polygons Circles in Unified Geometry Three Geometries Parallelism in Unified Geometry and the Influence of α Elliptic Geometry: Angle-Sum Theorem Pole-Polar Theory for Elliptic Geometry Angle Measure and Distance Related: Archimedes'' Method Hyperbolic Geometry: Angle-Sum Theorem A Concept for Area: AAA Congruence Parallelism in Hyperbolic Geometry Asymptotic Triangles in Hyperbolic Geometry Euclidean Geometry: Angle-Sum Theorem Median of a Trapezoid in Euclidean Geometry Similar Triangles in Euclidean Geometry Pythagorean Theorem Inequalities for Quadrilaterals: Unified Trigonometry An Inequality Concept for Unified Geometry Ratio Inequalities for Trapezoids Ratio Inequalities for Right Triangles Orthogonal Projection and "Similar" Triangles in Unified Geometry Unified Trigonometry: The Functions c(θ) and s(θ) Trigonometric Identities Classical Forms for c(θ) and s(θ) Lambert Quadrilaterals and the Function C(u) Identities for C(u) Classical Forms for C(u) The Pythagorean Relation for Unified Geometry Classical Unified Trigonometry Beyond Euclid: Modern Geometry Directed Distance: Stewart''s Theorem and the Cevian Formula Formulas for Special Cevians Circles: Power Theorems and Inscribed Angles Using Circles in Geometry Cross Ratio and Harmonic Conjugates The Theorems of Ceva and Menelaus Families of Mutually Orthogonal Circles Transformations in Modern Geometry Projective Transformations Affine Transformations Similitudes and Isometries Line Reflections: Building Blocks for Isometries and Similitudes Translations and Rotations Circular Inversion Non-Euclidean Geometry: Analytical Approach Law of Sines and Cosines for Unified Geometry Unifying Identities for Unified Trigonometry Half-Angle Identities for Unified Geometry The Shape of a Triangle in Unified Geometry: Cosine Inequality The Formulas of Gauss: Area of a Triangle Directed Distance: Theorems of Menelaus and Ceva Poincarè''s Model for Hyperbolic Geometry Other Models: Surface Theory Hyperbolic Parallelism and Bolyai''s Ideal Points Appendix A: Sketchpad Experiments Appendix B: Intuitive Spherical Geometry Appendix C: Proof in Geometry Appendix D: The Real Numbers and Least Upper Bound Appendix E: Floating Triangles/Quadrilaterals Appendix F: Axiom Systems for Geometry Solutions to Selected Problems Bibliography Index A Metric for City Centers The SAS Postulate and the ASA and SSS Theorems Euclid''s Superposition Proof: An Alternative to Axiom 12 Locus, Perpendicular Bisectors, and Symmetry The Exterior Angle Inequality Inequalities for Triangles Further Congruence Criteria Special Segments Associated with Triangles Quadrilaterals, Polygons, and Circles Quadrilaterals Congruence Theorems for Convex Quadrilaterals The Quadrilaterals of Saccheri and Lambert Polygons Circles in Unified Geometry Three Geometries Parallelism in Unified Geometry and the Influence of α Elliptic Geometry: Angle-Sum Theorem Pole-Polar Theory for Elliptic Geometry Angle Measure and Distance Related: Archimedes'' Method Hyperbolic Geometry: Angle-Sum Theorem A Concept for Area: AAA Congruence Parallelism in Hyperbolic Geometry Asymptotic Triangles in Hyperbolic Geometry Euclidean Geometry: Angle-Sum Theorem Median of a Trapezoid in Euclidean Geometry Similar Triangles in Euclidean Geometry Pythagorean Theorem Inequalities for Quadrilaterals: Unified Trigonometry An Inequality Concept for Unified Geometry Ratio Inequalities for Trapezoids Ratio Inequalities for Right Triangles Orthogonal Projection and "Similar" Triangles in Unified Geometry Unified Trigonometry: The Functions c(θ) and s(θ) Trigonometric Identities Classical Forms for c(θ) and s(θ) Lambert Quadrilaterals and the Function C(u) Identities for C(u) Classical Forms for C(u) The Pythagorean Relation for Unified Geometry Classical Unified Trigonometry Beyond Euclid: Modern Geometry Directed Distance: Stewart''s Theorem and the Cevian Formula Formulas for Special Cevians Circles: Power Theorems and Inscribed Angles Using Circles in Geometry Cross Ratio and Harmonic Conjugates The Theorems of Ceva and Menelaus Families of Mutually Orthogonal Circles Transformations in Modern Geometry Projective Transformations Affine Transformations Similitudes and Isometries Line Reflections: Building Blocks for Isometries and Similitudes Translations and Rotations Circular Inversion Non-Euclidean Geometry: Analytical Approach Law of Sines and Cosines for Unified Geometry Unifying Identities for Unified Trigonometry Half-Angle Identities for Unified Geometry The Shape of a Triangle in Unified Geometry: Cosine Inequality The Formulas of Gauss: Area of a Triangle Directed Distance: Theorems of Menelaus and Ceva Poincarè''s Model for Hyperbolic Geometry Other Models: Surface Theory Hyperbolic Parallelism and Bolyai''s Ideal Points Appendix A: Sketchpad Experiments Appendix B: Intuitive Spherical Geometry Appendix C: Proof in Geometry Appendix D: The Real Numbers and Least Upper Bound Appendix E: Floating Triangles/Quadrilaterals Appendix F: Axiom Systems for Geometry Solutions to Selected Problems Bibliography Index od Hyperbolic Geometry: Angle-Sum Theorem A Concept for Area: AAA Congruence Parallelism in Hyperbolic Geometry Asymptotic Triangles in Hyperbolic Geometry Euclidean Geometry: Angle-Sum Theorem Median of a Trapezoid in Euclidean Geometry Similar Triangles in Euclidean Geometry Pythagorean Theorem Inequalities for Quadrilaterals: Unified Trigonometry An Inequality Concept for Unified Geometry Ratio Inequalities for Trapezoids Ratio Inequalities for Right Triangles Orthogonal Projection and "Similar" Triangles in Unified Geometry Unified Trigonometry: The Functions c(θ) and s(θ) Trigonometric Identities Classical Forms for c(θ) and s(θ) Lambert Quadrilaterals and the Function C(u) Identities for C(u) Classical Forms for C(u) The Pythagorean Relation for Unified Geometry Classical Unified Trigonometry Beyond Euclid: Modern Geometry Directed Distance: Stewart''s Theorem and the Cevian Formula Formulas for Special Cevians Circles: Power Theorems and Inscribed Angles Using Circles in Geometry Cross Ratio and Harmonic Conjugates The Theorems of Ceva and Menelaus Families of Mutually Orthogonal Circles Transformations in Modern Geometry Projective Transformations Affine Transformations Similitudes and Isometries Line Reflections: Building Blocks for Isometries and Similitudes Translations and Rotations Circular Inversion Non-Euclidean Geometry: Analytical Approach Law of Sines and Cosines for Unified Geometry Unifying Identities for Unified Trigonometry Half-Angle Identities for Unified Geometry The Shape of a Triangle in Unified Geometry: Cosine Inequality The Formulas of Gauss: Area of a Triangle Directed Distance: Theorems of Menelaus and Ceva Poincarè''s Model for Hyperbolic Geometry Other Models: Surface Theory Hyperbolic Parallelism and Bolyai''s Ideal Points Appendix A: Sketchpad Experiments Appendix B: Intuitive Spherical Geometry Appendix C: Proof in Geometry Appendix D: The Real Numbers and Least Upper Bound Appendix E: Floating Triangles/Quadrilaterals Appendix F: Axiom Systems for Geometry Solutions to Selected Problems Bibliography Index Forms for C(u) The Pythagorean Relation for Unified Geometry Classical Unified Trigonometry Beyond Euclid: Modern Geometry Directed Distance: Stewart''s Theorem and the Cevian Formula Formulas for Special Cevians Circles: Power Theorems and Inscribed.