Charming Proofs : A Journey into Elegant Mathematics.

cover -- copyright page -- title page -- Preface -- Contents -- Introduction -- CHAPTER 1 A Garden of Integers -- 1.1 Figurate numbers -- 1.2 Sums of squares, triangular numbers, and cubes -- 1.3 There are infinitely many primes -- 1.4 Fibonacci numbers -- 1.5 Fermat's theorem -- 1.6 Wilson's theorem -- 1.7 Perfect numbers -- 1.8 Challenges -- CHAPTER 2 Distinguished Numbers -- 2.1 The irrationality of sqrt2 -- 2.2 The irrationality of sqrt{k} for non-square k -- 2.3 The golden ratio -- 2.4 pi and the circle -- 2.5 The irrationality of pi -- 2.6 The Comte de Buffon and his needle -- 2.7 e as a limit -- 2.8 An infinite series for e -- 2.9 The irrationality of e -- 2.10 Steiner's problem on the number e -- 2.11 The Euler-Mascheroni constant -- 2.12 Exponents, rational and irrational -- 2.13 Challenges -- CHAPTER 3 Points in the Plane -- 3.1 Pick's theorem -- 3.2 Circles and sums of two squares -- 3.3 The Sylvester-Gallai theorem -- 3.4 Bisecting a set of 100,000 points -- 3.5 Pigeons and pigeonholes -- 3.6 Assigning numbers to points in the plane -- 3.7 Challenges -- CHAPTER 4 The Polygonal Playground -- 4.1 Polygonal combinatorics -- 4.2 Drawing an n-gon with given side lengths -- 4.3 The theorems of Maekawa and Kawasaki -- 4.4 Squaring polygons -- 4.5 The stars of the polygonal playground -- 4.6 Guards in art galleries -- 4.7 Triangulation of convex polygons -- 4.8 Cycloids, cyclogons, and polygonal cycloids -- 4.9 Challenges -- CHAPTER 5 A Treasury of TriangleTheorems -- 5.1 The Pythagorean theorem -- 5.2 Pythagorean relatives -- 5.3 The inradius of a right triangle -- 5.4 Pappus' generalization of the Pythagorean theorem -- 5.5 The incircle and Heron's formula -- 5.6 The circumcircle and Euler's triangle inequality -- 5.7 The orthic triangle -- 5.8 The Erdos-Mordell inequality -- 5.9 The Steiner-Lehmus theorem.

5.10 The medians of a triangle -- 5.11 Are most triangles obtuse? -- 5.12 Challenges -- CHAPTER 6 The Enchantment of the Equilateral Triangle -- 6.1 Pythagorean-like theorems -- 6.2 The Fermat point of a triangle -- 6.3 Viviani's theorem -- 6.4 A triangular tiling of the plane and Weitzenböck's inequality -- 6.5 Napoleon's theorem -- 6.6 Morley's miracle -- 6.7 Van Schooten's theorem -- 6.8 The equilateral triangle and the golden ratio -- 6.9 Challenges -- CHAPTER 7 The Quadrilaterals' Corner -- 7.1 Midpoints in quadrilaterals -- 7.2 Cyclic quadrilaterals -- 7.3 Quadrilateral equalities and inequalities -- 7.4 Tangential and bicentric quadrilaterals -- 7.5 Anne's and Newton's theorems -- 7.6 Pythagoras with a parallelogram andequilateral triangles -- 7.7 Challenges -- CHAPTER 8 Squares Everywhere -- 8.1 One-square theorems -- 8.2 Two-square theorems -- 8.3 Three-square theorems -- 8.4 Four and more squares -- 8.5 Squares in recreational mathematics -- 8.6 Challenges -- CHAPTER 9 Curves Ahead -- 9.1 Squarable lunes -- 9.2 The amazing Archimedean spiral -- 9.3 The quadratrix of Hippias -- 9.4 The shoemaker's knife and the salt cellar -- 9.5 The Quetelet and Dandelin approach to conics -- 9.6 Archimedes triangles -- 9.7 Helices -- 9.8 Challenges -- CHAPTER 10 Adventures in Tilingand Coloring -- 10.1 Plane tilings and tessellations -- 10.2 Tiling with triangles and quadrilaterals -- 10.3 Infinitely many proofs of the Pythagorean theorem -- 10.4 The leaping frog -- 10.5 The seven friezes -- 10.6 Colorful proofs -- 10.7 Dodecahedra and Hamiltonian circuits -- 10.8 Challenges -- CHAPTER 11 Geometry in Three Dimensions -- 11.1 The Pythagorean theorem in three dimensions -- 11.2 Partitioning space with planes -- 11.3 Corresponding triangles on three lines -- 11.4 An angle-trisecting cone -- 11.5 The intersection of three spheres.

11.6 The fourth circle -- 11.7 The area of a spherical triangle -- 11.8 Euler's polyhedral formula -- 11.9 Faces and vertices in convex polyhedra -- 11.10 Why some types of faces repeat in polyhedra -- 11.11 Euler and Descartes a la Polya -- 11.12 Squaring squares and cubing cubes -- 11.13 Challenges -- CHAPTER 12 Additional Theorems, Problems, and Proofs -- 12.1 Denumerable and nondenumerble sets -- 12.2 The Cantor-Schro¨ der-Bernstein theorem -- 12.3 The Cauchy-Schwarz inequality -- 12.4 The arithmetic mean-geometric mean inequality -- 12.5 Two pearls of origami -- 12.6 How to draw a straight line -- 12.7 Some gems in functional equations -- 12.8 Functional inequalities -- 12.9 Euler's series for pi^2/6 -- 12.10 The Wallis product -- 12.11 Stirling's approximation of n! -- 12.12 Challenges -- Solutions to the Challenges -- Chapter 1 -- Chapter 2 -- Chapter 3 -- Chapter 4 -- Chapter 5 -- Chapter 6 -- Chapter 7 -- Chapter 8 -- Chapter 9 -- Chapter 10 -- Chapter 11 -- Chapter 12 -- References -- Index -- About the Authors.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.