Proof in Geometry : With Mistakes in Geometric Proofs

Introduction 1(2) 1. First student's question 1(1) 2. Second student's question 1(1) 3. Third student's question 2(1) 4. How to find the answers 2(1) CHAPTER 1. What Is a Proof? 3(4) 5. Induction and deduction 3(2) 6. Application to geometry 5(2) CHAPTER 2. Why Are Proofs Necessary? 7(6) 7. The law of sufficient reason 7(1) 8. Dangers of "obviousness" 7(3) 9. Dangers of particular cases 10(1) 10. Geometry as a scientific system 11(1) 11. Summary 12(1) CHAPTER 3. How Should a Proof Be Constructed? 13(24) 12. Correct reasoning 13(2) 13. Incorrect reasoning 15(1) 14. Converse theorems 16(2) 15. Distinguishing between direct and converse theorems 18(1) 16. Conditional and categorical statements 19(1) 17. Avoiding particular cases 20(3) 18. Incomplete proofs 23(2) 19. Circular reasoning 25(2) 20. Requirements for a correct proof 27(1) 21. How to find a correct proof 28(1) 22. Analysis 29(3) 23. Synthesis 32(1) 24. Direct and indirect proofs 33(4) CHAPTER 4. What Propositions in Geometry Are Accepted without Proof? 37 25. Bases for selection of axioms 37(1) 26. Properties of a system of axioms 38(1) 27. Analogy from algebra 39(2) 28. Axioms of connection 41(1) 29. Axioms of order 42(4) 30. Axioms of congruence 46(2) 31. Axioms of continuity 48(4) 32. Theorems based on the axioms of continuity 52(2) 33. Axiom of parallelism 54(1) 34. Reduction of the number of axioms 54(1) 35. Summary 55