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 Erd s: 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.
Author(s): Claudi Alsina, Roger B. Nelsen
Series: Dolciani Mathematical Expositions 55
Publisher: American Mathematical Society
Year: 2020
Language: English
Pages: 304
Cover
Title Page
Copyright Page
Table of Contents
Preface
Chapter 1. Simple Quadrilaterals
1.1. Introduction
1.2. The Varignon parallelogram
1.3. The quadrilateral law
1.4. Diagonal midpoints, the Newton line, and Anne's theorem
1.5. Area formulas and inequalities
1.6. Van Aubel's theorem
1.7. Equilic quadrilaterals
1.8. Challenges
Chapter 2. Quadrilaterals and Their Circles
2.1. Introduction
2.2. Cyclic quadrilaterals
2.3. Ptolemy's theorem and its consequences
2.4. The diagonals of a cyclic quadrilateral
2.5. Brahmagupta's formula
2.6. Maltitudes and the anticenter of a cyclic quadrilateral
2.7. Tangential quadrilaterals and Newton's theorem
2.8. Bicentric quadrilaterals
2.9. Extangential quadrilaterals and Urquhart's theorem
2.10. Challenges
Chapter 3. Diagonals of Quadrilaterals
3.1. Introduction
3.2. Orthodiagonal quadrilaterals
3.3. Equidiagonal quadrilaterals
3.4. Kites
3.5. Rhombi
3.6. Midsquare quadrilaterals
3.7. Summary
3.8. Challenges
Chapter 4. Properties of Trapezoids
4.1. Introduction
4.2. Pythagorean-like theorems for trapezoids
4.3. Trapezoid area and the bimedians
4.4. Trapezoid diagonals
4.5. Isosceles trapezoids
4.6. Trilateral trapezoids
4.7. Right trapezoids
4.8. Challenges
Chapter 5. Applications of Trapezoids
5.1. Trapezoidal means
5.2. Trapezoids and the Hermite-Hadamard inequality
5.3. Trapezoidal reptiles and infinite series
5.4. Fibonacci trapezoids
S.S. Right trapezoids and the Erdos-Mordell inequality
5.6. Challenges
Chapter 6. Garfield Trapezoids and Rectangles
6.1. Introduction
6.2. Inequalities for means
6.3. Diophantus of Alexandria's sum of squares identity and the Cauchy-Schwarz inequality
6.4. Trigonometric identities
6.5. Arctangent identities
6.6. Challenges
Chapter 7. Parallelograms
7.1. Introduction
7.2. Some basic parallelogram theorems
7.3. Papp us' s area theorem
7.4. Bhaskara and parallelograms
7.5. The area of a parallelogram as a determinant
7.6. Parallelograms in space
7.7. The mediant property and Simpson's paradox
7.8. Challenges
Chapter 8. Rectangles
8.1. Introduction
8.2. Characterizations
8.3. Reciprocal rectangles
8.4. Golden rectangles
8.5. Silver and other metallic rectangles
8.6. Fibonacci rectangles
8.7. Challenges
Chapter 9. Squares
9.1. Introduction
9.2. Characterizations
9.3. The bride's chair and the Vecten configuration
9.4. Inscribing squares in triangles
9.5. Squared squares and related objects
9.6. Pythagorean triples
9.7. Challenges
Chapter 10. Special Quadrilaterals
10.1. Introduction
10.2. Concave quadrilaterals
10.3. Complex quadrilaterals
10.4. Skew quadrilaterals
10.5. Saccheri and Lambert quadrilaterals
10.6. Challenges
Chapter 11.Quadrilateral Numbers
11.1. Introduction
11.2. Square and oblong numbers
11.3. Square sums of two consecutive square numbers
11.4. Trapezoidal and polite numbers
11.5. Challenges
Solutions to the Challenges
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 1O
Chapter 11
Appendix: A Quadrilateral Glossary
Credits and Permissions
Bibliography
Index
