1:46 Gerard A. Venema Foundations of Geometry 2011 PDF - 3 MB 34 Chapter 2 Axiomatic Systems and Incidence Geometry Here are several other theorems from incidence geometry. You can practice what you have learned in this chapter by writing proofs for them. Theorem 2.6.3. If l is any line, then there exists at least one point P such that P does not lie оn e. Theorem 2.6.4. If P is any point, then there are at least two distinct lines l and m such that P lies on both l and m. Theorem 2.6.5. If l is any line, then there exist lines m and n such that l, m, and n are distinct and both m and n intersect l. Theorem 2.6.6. If P is any point, then there exists at least one line l such that P does not lie оn e. Theorem 2.6.7. There exist three distinct lines such that no point lies on all three of the lines. Theorem 2.6.8. If P is any point, then there exist points Q and R such that P, Q, and R are noncollinear. Theorem 2.6.9. If P and Q are two points such that P + Q, then there exists a point R such that P, Q, and R are noncollinear. Every theorem has a proper context and that context is an axiomatic system. Thus every theorem has unstated hypotheses, namely that certain axioms are assumed true. For example, the theorems above are theorems in incidence geometry. This means that every one of them includes the unstated hypothesis that the three incidence axioms are assumed true. In the case of Theorem 2.6.7, the unstated hypotheses are the only hypotheses. One final remark about writing proofs: Except for the gaps we discussed in Chapter 1, Euclid's proofs serve as excellent models for you to follow. Euclid usually includes just the right amount of detail and clearly states his reasons for each step in exactly the way that is advocated in this chapter. He also includes helpful explanations of where the proof is going, so that the reader has a better chance of understanding the big picture. In learning to write good proofs you can do no better than to study Euclid's proofs, especially those from Book I of the Elements. As you come to master those proofs you will begin to appreciate them more and more. You will eventually find yourself reading and enjoying not just the proofs themselves, but also Heath’s commentary [22] on the proofs. Heath often explains why Euclid did things as he did and also indicates how other geometers have proved the same theorem. Of course Euclid uses language quite differently from the way we do. His theorem statements themselves do not serve as good models of the careful statements that modern standards of rigor demand. EXERCISES 2.6 1. Prove the converse to Theorem 2.6.2. 2. Prove Theorem 2.6.3. 3. Prove Theorem 2.6.4. 4. Prove Theorem 2.6.5. 5. Prove Theorem 2.6.6. 6. Prove Theorem 2.6.7. 7. Prove Theorem 2.6.8. 8. Prove Theorem 2.6.9. СНАРТER 3