College Geometry with GeoGebra

From two authors who embrace technology in the classroom and value the role of collaborative learning comes College Geometry Using GeoGebra, a book that is ideal for geometry courses for both mathematics and math education majors. The book's discovery-based approach guides students to explore geometric worlds through computer-based activities, enabling students to make observations, develop conjectures, and write mathematical proofs. This unique textbook helps students understand the underlying concepts of geometry while learning to use GeoGebra software—constructing various geometric figures and investigating their properties, relationships, and interactions. The text allows students to gradually build upon their knowledge as they move from fundamental concepts of circle and triangle geometry to more advanced topics such as isometries and matrices, symmetry in the plane, and hyperbolic and projective geometry.

Emphasizing active collaborative learning, the text contains numerous fully-integrated computer lab activities that visualize difficult geometric concepts and facilitate both small-group and whole-class discussions. Each chapter begins with engaging activities that draw students into the subject matter, followed by detailed discussions that solidify the student conjectures made in the activities and exercises that test comprehension of the material. Written to support students and instructors in active-learning classrooms that incorporate computer technology, College Geometry with GeoGebra is an ideal resource for geometry courses for both mathematics and math education majors.

Preface

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