Description
Book SynopsisTable of ContentsPreface
Especially for Students
Notes for Instructors
Our Motivation, Philosophy, and Pedagogy
Prerequisites and Chapter Dependencies
Acknowledgments
ONEUsing GeoGebra
1.1 Activities: Getting Started with GeoGebra
1.2 Discussion: Exploring and Conjecturing
Some GeoGebra Tips
Constructing −→ Exploring −→ Conjecturing:
Inductive Reasoning
Language of Geometry
Explorations, Observations, Questions
The Family of Quadrilaterals
Angles Inscribed in Circles
Rules of Logic
1.3 Exercises
1.4 Chapter Overview
TWO Constructing → Proving
2.1 Activities
2.2 Discussion: Euclid’s Postulates and Constructions
Euclid’s Postulates
Congruence and Similarity
Constructions
Geometric Language Revisited
Conditional Statements: Implication
Using Robust Constructions to Develop a Proof
Angles and Measuring Angles
Constructing Perpendicular and Parallel Lines
Properties of Triangles
Euclid’s Parallel Postulate
Euclid’s Constructions in the Elements
Ideas About Betweenness
2.3 Exercises
2.4 Chapter Overview
THREE Mathematical Arguments and Triangle Geometry
3.1 Activities
3.2 Discussion
Deductive Reasoning
Universal and Existential Quantifiers
Negating a Quantified Statement
Direct Proof and Disproof by Counterexample
Step-by-Step Proofs
Congruence Criteria for Triangles
The Converse and the Contrapositive
Concurrence Properties for Triangles
Ceva’s Theorem and Its Converse
Brief Excursion into Circle Geometry
The Circumcircle of ΔABC
The Nine-Point Circle: A First Pass
Menelaus’ Theorem and Its Converse
3.3 Exercises
3.4 Chapter Overview
FOUR Circle Geometry and Proofs
4.1 Activities
4.2 Discussion
Axiom Systems: Ancient and Modern Approaches
Language of Circles
Inscribed Angles
Mathematical Arguments
Additional Methods of Proof
Cyclic Quadrilaterals
Incircles and Excircles
Some Interesting Families of Circles
The Arbelos and the Salinon
Power of a Point
The Radical Axis
The Nine-Point Circle: A Second Pass
4.3 Exercises
4.4 Chapter Overview
FIVE Analytic Geometry
5.1 Activities
5.2 Discussion
Points
Lines
Distance
Using Coordinates in Proofs
Another Look at the Radical Axis
Polar Coordinates
The Nine-Point Circle, Revisited
5.3 Exercises
5.4 Chapter Overview
SIX Taxicab Geometry
6.1 Activities
6.2 Discussion
An Axiom System for Metric Geometry
Circles
Ellipses
Measuring Distance from a Point to a Line
Parabolas
Hyperbolas
Axiom Systems
6.3 Exercises
6.4 Chapter Overview
SEVEN Finite Geometries
7.1 Activities
7.2 Discussion
An Axiom System for an Affine Plane
An Axiom System for a Projective Plane
Duality
Relating Affine Planes to Projective Planes
Coordinates for Finite Geometries
7.3 Exercises
7.4 Chapter Overview
EIGHTTransformational Geometry
8.1 Activities
8.2 Discussion
Transformations
Isometries
Other Transformations
Composition of Isometries
Inverse Isometries
Using Isometries in Proofs
Isometries in Space
8.3 Exercises
8.4 Chapter Overview
NINE Isometries and Matrices
9.1 Activities
9.2 Discussion
Using Vectors to Represent Translations
Using Matrices to Represent Rotations
Using Matrices to Represent Reflections
Composition of Isometries
The General Form of a Matrix Representation
Using Matrices in Proofs
Similarity Transformations
9.3 Exercises
9.4 Chapter Overview
TENSymmetry in the Plane
10.1 Activities
10.2 Discussion
Symmetries
Groups of Symmetries
Classifying Figures by Their Symmetries
Friezes and Symmetry
Wallpaper Symmetry
Tilings
10.3 Exercises
10.4 Chapter Overview
ELEVEN Hyperbolic Geometry
Part I: Exploring a New Universe
11.1 Activities Part I
11.2 Discussion Part I
Hyperbolic Lines and Segments
The Poincaré Disk Model of the Hyperbolic Plane
Measuring Distance in the Poincaré Disk Model
Hyperbolic Circles
Hyperbolic Triangles
Circumcircles and Incircles of Hyperbolic Triangles
Congruence of Triangles in the Hyperbolic Plane
Part II: The Parallel Postulate in Hyperbolic Geometry
11.3 Activities Part II
11.4 Discussion Part II
The Hyperbolic and Elliptic Parallel Postulates
The Angle of Parallelism
The Exterior Angle Theorem
Quadrilaterals in the Hyperbolic Plane
Another Look at Triangles in the Hyperbolic Plane
Area in the Hyperbolic Plane
11.5 Exercises
The Upper-Half-Plane Model
11.6 Chapter Overview
TWELVE Projective Geometry
12.1 Activities
12.2 Discussion
An Axiom System
Models for the Projective Plane
Duality
Coordinates for Projective Geometry
Projective Transformations
12.3 Exercises
12.4 Chapter Overview
APPENDIX A Trigonometry
A.1 Activities
A.2 Discussion
Right Triangle Trigonometry
Unit Circle Trigonometry
Solving Trigonometric Equations
Double Angle Formulas
Angle Sum Formulas
Half-Angle Formulas
The Law of Sines and the Law of Cosines
A.3 Exercises
APPENDIX B Calculating with Matrices
B.1 Activities
B.2 Discussion
Linear Combinations of Vectors
Dot Product of Vectors
Multiplying a Matrix Times a Vector
Multiplying Two Matrices
The Determinant of a Matrix
B.3 Exercises
BIBLIOGRAPHY
INDEX
