**Classroom Resource Materials**

Volume: 52;
2015;
187 pp;
Softcover

**Print ISBN: 978-0-88385-790-8
Product Code: CLRM/52**

List Price: $50.00

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**Electronic ISBN: 978-1-61444-121-2
Product Code: CLRM/52.E**

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# Proofs Without Words III: Further Exercises in Visual Thinking

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*Roger B. Nelsen*

MAA Press: An Imprint of the American Mathematical Society

Proofs without words (PWWs) are figures or diagrams that help the reader see why a particular mathematical statement is true, and how one might begin to formally prove it true. PWWs are not new, many date back to classical Greece, ancient China, and medieval Europe and the Middle East. PWWs have been regular features of the MAA journals Mathematics Magazine and The College Mathematics Journal for many years, and the MAA published the collections of PWWs Proofs Without Words: Exercises in Visual Thinking in 1993 and Proofs Without Words II: More Exercises in Visual Thinking in 2000. This book is the third such collection of PWWs.

#### Reviews & Endorsements

[This book] is a hugely enjoyable read, with intrigue and interest in every diagram.

-- Nick Lord, The Mathematical Gazette

# Table of Contents

## Proofs Without Words III: Further Exercises in Visual Thinking

- Cover cov11
- Half title i3
- Copyright ii4
- Title iii5
- Series iv6
- Introduction vii9
- Contents ix11
- Geometry & Algebra 117
- The Pythagorean Theorem XIII 319
- The Pythagorean Theorem XIV 420
- The Pythagorean Theorem XV 521
- The Pythagorean Theorem XVI 622
- Pappus' Generalization of the Pythagorean Theorem 723
- A Reciprocal Pythagorean Theorem 824
- A Pythagorean-Like Formula 925
- Four Pythagorean-Like Theorems 1026
- Pythagoras for a Right Trapezoid 1430
- Pythagoras for a Clipped Rectangle 1531
- Heron's Formula 1632
- Every Triangle Has Infinitely Many Inscribed Equilateral Triangles 1733
- Every Triangle Can Be Dissected into Six Isosceles Triangles 1834
- More Isosceles Dissections 1935
- Viviani's Theorem II 2036
- Viviani's Theorem III 2137
- Ptolemy's Theorem I 2238
- Ptolemy's Theorem II 2339
- Equal Areas in a Partition of a Parallelogram 2440
- The Area of an Inner Square 2541
- The Parallelogram Law 2642
- The Length of a Triangle Median via the Parallelogram Law 2743
- Two Squares and Two Triangles 2844
- The Inradius of an Equilateral Triangle 2945
- A Line Through the Incenter of a Triangle 3046
- The Area and Circumradius of a Triangle 3147
- Beyond Extriangles 3248
- A 45° Angle Sum 3349
- Trisection of a Line Segment II 3450
- Two Squares with Constant Area 3551
- Four Squares with Constant Area 3652
- Squares in Circles and Semicircles 3854
- The Christmas Tree Problem 3955
- The Area of an Arbelos 4056
- The Area of a Salinon 4157
- The Area of a Right Triangle 4258
- The Area of a Regular Dodecagon II 4359
- Four Lunes Equal One Square 4460
- Lunes and the Regular Hexagon 4561
- The Volume of a Triangular Pyramid 4662
- Algebraic Areas IV 4763
- Componendo et Dividendo, a Theorem on Proportions 4864
- Completing the Square II 4965
- Candido's Identity 5066

- Trigonometry, Calculus, & Analytic Geometry 5167
- Sine of a Sum or Difference (via the Law of Sines) 5369
- Cosine of the Difference I 5470
- Sine of the Sum IV and Cosine of the Difference II 5571
- The Double Angle Formulas IV 5672
- Euler's Half Angle Tangent Formula 5773
- The Triple Angle Sine and Cosine Formulas I 5874
- The Triple Angle Sine and Cosine Formulas II 5975
- Trigonometric Functions of 15° and 75° 6076
- Trigonometric Functions of Multiples of 18° 6177
- Mollweide's Equation II 6278
- Newton's Formula (for the General Triangle) 6379
- A Sine Identity for Triangles 6480
- Cofunction Sums 6581
- The Law of Tangents I 6682
- The Law of Tangents II 6783
- Need a Solution to x + y = xy? 6884
- An Identity for sec x + tan x 6985
- A Sum of Tangent Products 7086
- A Sum and Product of Three Tangents 7187
- A Product of Tangents 7288
- Sums of Arctangents II 7389
- One Figure, Five Arctangent Identities 7490
- The Formulas of Hutton and Strassnitzky 7591
- An Arctangent Identity 7692
- Euler's Arctangent Identity 7793
- Extrema of the Function a cos t+ b sin t 7894
- A Minimum Area Problem 7995
- The Derivative of the Sine 8096
- The Derivative of the Tangent 8197
- Geometric Evaluation of a Limit II 8298
- The Logarithm of a Number and Its Reciprocal 8399
- Regions Bounded by the Unit Hyperbolawith Equal Area 84100
- The Weierstrass Substitution II 85101
- Look Ma, No Substitution! 86102
- Integrating the Natural Logarithm 87103
- The Integrals of cos[sup(2)]θ and sec[sup(2)]θ 88104
- A Partial Fraction Decomposition 89105
- An Integral Transform 90106

- Inequalities 91107
- The Arithmetic Mean–Geometric Mean Inequality VII 93109
- The Arithmetic Mean–Geometric Mean Inequality VIII (via Trigonometry) 94110
- The Arithmetic Mean-Root Mean Square Inequality 95111
- The Cauchy-Schwarz Inequality II (via Pappus' theorem) 96112
- The Cauchy-Schwarz Inequality III 97113
- The Cauchy-Schwarz Inequality IV 98114
- The Cauchy-Schwarz Inequality V 99115
- Inequalities for the Radii of Right Triangles 100116
- Ptolemy's Inequality 101117
- An Algebraic Inequality I 102118
- An Algebraic Inequality II 103119
- The Sine is Subadditive on [0,π] 104120
- The Tangent is Superadditive on [0,π/2) 105121
- Inequalities for Two Numbers whose Sum is One 106122
- Padoa's Inequality 107123
- Steiner's Problem on the Number e 108124
- Simpson's Paradox 109125
- Markov's Inequality 110126

- Integers & Integer Sums 111127
- Sums of Odd Integers IV 113129
- Sums of Odd Integers V 114130
- Alternating Sums of Odd Numbers 115131
- Sums of Squares X 116132
- Sums of Squares XI 117133
- Alternating Sums of Consecutive Squares 118134
- Alternating Sums of Squares of Odd Numbers 119135
- Archimedes' Sum of Squares Formula 120136
- Summing Squares by Counting Triangles 121137
- Squares Modulo 3 122138
- The Sum of Factorials of Order Two 123139
- The Cube as a Double Sum 124140
- The Cube as an Arithmetic Sum II 125141
- Sums of Cubes VIII 126142
- The Difference of Consecutive Integer Cubes is Congruent to 1 Modulo 6 127143
- Fibonacci Identities II 128144
- Fibonacci Tiles 129145
- Fibonacci Trapezoids 130146
- Fibonacci Triangles and Trapezoids 131147
- Fibonacci Squares and Cubes 132148
- Every Octagonal Number is the Differenceof Two Squares 133149
- Powers of Two 134150
- Sums of Powers of Four 135151
- Sums of Consecutive Powers of n via Self-Similarity 136152
- Every Fourth Power Greater than One is the Sum of Two Non-consecutive Triangular Numbers 137153
- Sums of Triangular Numbers V 138154
- Alternating Sums of Triangular Numbers II 139155
- Runs of Triangular Numbers 140156
- Sums of Every Third Triangular Number 141157
- Triangular Sums of Odd Numbers 142158
- Triangular Numbers are Binomial Coefficients 143159
- The Inclusion-Exclusion Formula for Triangular Numbers 144160
- Partitioning Triangular Numbers 145161
- A Triangular Identity II 146162
- A Triangular Sum 147163
- A Weighted Sum of Triangular Numbers 148164
- Centered Triangular Numbers 149165
- Jacobsthal Numbers 150166

- Infinite Series & Other Topics 151167
- Geometric Series V 153169
- Geometric Series VI 154170
- Geometric Series VII (via Right Triangles) 155171
- Geometric Series VIII 156172
- Geometric Series IX 157173
- Differentiated Geometric Series II 158174
- A Geometric Telescope 159175
- An Alternating Series II 160176
- An Alternating Series III 161177
- The Alternating Series Test 162178
- The Alternating Harmonic Series II 163179
- Galileo's Ratios II 164180
- Slicing Kites Into Circular Sectors 165181
- Nonnegative Integer Solutions and Triangular Numbers 166182
- Dividing a Cake 167183
- The Number of Unordered Selections with Repetitions 168184
- A Putnam Proof Without Words 169185
- On Pythagorean Triples 170186
- Pythagorean Quadruples 171187
- The Irrationality of √2 172188
- Z × Z is a Countable Set 173189
- A Graph Theoretic Summation of the First n Integers 174190
- A Graph Theoretic Decomposition of Binomial Coefficients 175191
- (0,1) and [0,1] Have the Same Cardinality 176192
- A Fixed Point Theorem 177193
- In Space, Four Colors are not Enough 178194

- Sources 179195
- Index of Names 185201
- About the Author 187203