**Dolciani Mathematical Expositions**

Volume: 55;
2020;
304 pp;
Hardcover

MSC: Primary 51;

**Print ISBN: 978-1-4704-5312-1
Product Code: DOL/55**

List Price: $59.00

AMS Member Price: $44.25

MAA Member Price: $44.25

**Electronic ISBN: 978-1-4704-5465-4
Product Code: DOL/55.E**

List Price: $59.00

AMS Member Price: $44.25

MAA Member Price: $44.25

#### Supplemental Materials

# A Cornucopia of Quadrilaterals

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*Claudi Alsina; Roger B. Nelsen*

MAA Press: An Imprint of the American Mathematical Society

A Cornucopia of Quadrilaterals collects and organizes
hundreds of beautiful and surprising results about four-sided
figures—for example, that the midpoints of the sides of any
quadrilateral are the vertices of a parallelogram, or that in a convex
quadrilateral (not a parallelogram) the line through the midpoints of
the diagonals (the Newton line) is equidistant from opposite
vertices, or that, if your quadrilateral has an inscribed circle, its
center lies on the Newton line. There are results dating back to
Euclid: the side-lengths of a pentagon, a hexagon, and a decagon
inscribed in a circle can be assembled into a right triangle (the
proof uses a quadrilateral and circumscribing circle); and results
dating to Erdős: from any point in a triangle the sum of the
distances to the vertices is at least twice as large as the sum of the
distances to the sides.

The book is suitable for serious study, but it equally rewards the
reader who dips in randomly. It contains hundreds of challenging
four-sided problems. Instructors of number theory, combinatorics,
analysis, and geometry will find examples and problems to enrich their
courses. The authors have carefully and skillfully organized the
presentation into a variety of themes so the chapters flow seamlessly
in a coherent narrative journey through the landscape of
quadrilaterals. The authors' exposition is beautifully clear and
compelling and is accessible to anyone with a high school background
in geometry.

#### Readership

Undergraduate and graduate students interested in geometry.

#### Table of Contents

# Table of Contents

## A Cornucopia of Quadrilaterals

- Cover Cover11
- Title Page iii4
- Contents v6
- Preface ix10
- Chapter 1. Simple Quadrilaterals 114
- Chapter 2. Quadrilaterals and Their Circles 3144
- 2.1. Introduction 3144
- 2.2. Cyclic quadrilaterals 3245
- 2.3. Ptolemy’s theorem and its consequences 3548
- 2.4. The diagonals of a cyclic quadrilateral 3952
- 2.5. Brahmagupta’s formula 4154
- 2.6. Maltitudes and the anticenter of a cyclic quadrilateral 4457
- 2.7. Tangential quadrilaterals and Newton’s theorem 4558
- 2.8. Bicentric quadrilaterals 4962
- 2.9. Extangential quadrilaterals and Urquhart’s theorem 5265
- 2.10. Challenges 5871

- Chapter 3. Diagonals of Quadrilaterals 6376
- Chapter 4. Properties of Trapezoids 89102
- Chapter 5. Applications of Trapezoids 107120
- Chapter 6. Garfield Trapezoids and Rectangles 127140
- Chapter 7. Parallelograms 143156
- 7.1. Introduction 143156
- 7.2. Some basic parallelogram theorems 145158
- 7.3. Pappus’s area theorem 148161
- 7.4. Bhāskara and parallelograms 149162
- 7.5. The area of a parallelogram as a determinant 151164
- 7.6. Parallelograms in space 154167
- 7.7. The mediant property and Simpson’s paradox 155168
- 7.8. Challenges 157170

- Chapter 8. Rectangles 161174
- Chapter 9. Squares 185198
- Chapter 10. Special Quadrilaterals 205218
- Chapter 11. Quadrilateral Numbers 221234
- Solutions to the Challenges 237250
- Appendix A: A Quadrilateral Glossary 273286
- Credits and Permissions 277290
- Bibliography 279292
- Index 285298
- Back Cover 292305