Introduction ⅲ

Chapter 0: Problem Book by Area

Algebra 2
Integer 3
Figure 3
Function 8
Case 9 · Probability

Chapter 1: Beautiful Theorems You Can Enjoy in Middle School Mathematics

1. Three proofs of the formula for the solution of a quadratic equation 14
2 Four Beautiful Proofs of the Pythagorean Theorem 17
3 Proofs for the Existence of Water Depth 21
25 Three Proofs of the Four-Line Theorem (Apollonius' Theorem)
5. Length of the diagonal of a regular pentagon and its construction method 29
6 Formula for the number of divisors and properties of square numbers 32
Four proofs showing that 7 is an irrational number 35
8 How to solve double roots? 38
9. How to find the number of cases of road direction 42
10 Two proofs that there are five types of regular polyhedra 46
11 Problem 50: Counting the Types of Tetris Blocks

Chapter 2: Slightly different formulas from those in the textbook

12 Handy Trigonometric Ratios to Remember 56
Example 59 Proving Heron's Formula
14 Proof and Example of Equilibrium Shift in Graph 64
15 Two Ways to Graph Exponential Functions 67
16 Three Ways to Solve Partial Fractions 71
17 Factoring Formulas (Difference of Squares, Sum) 75
78 Properties to Remember in 18-Complex Conjugate Numbers
19 Determination of Quadratic Functions and Background 81
Proof of the cosine theorem using the inner product of 20 vectors 85
21 Various triangle inequalities (absolute values, complex numbers, vectors) 89
22 Four Properties of Two-Variable Symmetry and Basic Symmetry 93
23 Integer solutions to first-order inequalities 97
24 Relationship between the radius of the circumcircle and the area of ​​a triangle 100

Chapter 3: Which is better, elegant proof or consistent proof?

Proof of the three formulas for the distance between a point and a straight line 104
26 Three Ways to Find the Equation of a Plane 109
27 Comparison of two methods for finding the angle between two straight lines 113
28 Cheba's Theorem: Three Proofs 118
29 Proof and Application of Ptolemy's Theorem 122
Proof and Example of Ptolemy's Inequality 127
31 Various proofs of inequalities 131
32 Five Proofs of Nesbitt's Inequality 134
33 Various proofs of numbers with a larger pi 138
34 The Fourth Power Theorem (Area and Orthogonal Projection of Figures) 143
35 A Beautiful Proof That There Are Infinite Prime Numbers 147
36 Three proofs that the interval of prime numbers has no maximum 150
37 Proof of Euler's Polyhedron Theorem 153

Chapter 4: Beautiful theorems and formulas hidden in familiar topics

38 How to calculate the probability of a draw in rock, paper, scissors and its value 158
39 Find out the probability that two people have the same birthday 161
40 Bankruptcy Probability and Ignition Formula 165
41 Optimal Strategy for Rock, Paper, Scissors and Tokyo University Entrance Exam Questions 169
42 Nim (Multiple Stone Grab Game) Winning Strategy 173
43 Derivation of the formula for parabolic projectiles and methods for increasing the distance 178
44. The Luo triangle and the regular figure 182
Introduction to the Four-Color Theorem and Proof of the Five-Color Theorem 185
46 Ramsey's Theorem and the Six-person Problem 189
47 Planar Graphs and Applications of Euler's Theorem 192

Chapter 5: Even difficult theorems and formulas are fun when their essence is revealed.

Formula for calculating the volume of a tetrahedron at 48 coordinates 198
49 Chebyshev polynomials 202
Proving that 50 is an irrational number 205
51 Formulas for the sum and product of the interior angles of a triangle 208
52 Three Proofs of the Fermat Point of a Triangle 213
53 Tent Ideas and Their Nature (Background of the University of Tokyo Entrance Examination) 218
54 Cartano's Formula and Examples 222
55 Fermat's Numbers and Properties 226
56 The meaning of Catalan numbers and their recurrence formulas 229
57 Probability and Proof Related to Polya's Urn 233
58 Proof of the number of warriors and the number of bells 236
59 Area and sum of interior angles of a triangle on a sphere 240
60 Frank-Morley Theorem and Proof 244

Index 248