**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**