4. Prove Theorem 2.6.5.
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Author:Erwin Kreyszig
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![1:44
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 wh
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 i
оn l.
Theorem 2.6.4. If P is any point, then there are at least two distinct lines l and m such th
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 a
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 i
оn l.
Theorem 2.6.7. There exist three distinct lines such that no point lies on all three of th
lines.
Theorem 2.6.8. If P is any point, then there exist points Q and R such that P, Q, and Ra
попcollinear.
Theorem 2.6.9. If P and Q are two points such that P + Q, then there exists a point R suc
that P, Q, and R are noncollinear.
Every theorem has a proper context and that context is an axiomatic system. Thi
every theorem has unstated hypotheses, namely that certain axioms are assumed true. F
example, the theorems above are theorems in incidence geometry. This means that eve:
one of them includes the unstated hypothesis that the three incidence axioms are assume
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
Euclid's proofs serve as excellent models for you to follow. Euclid usually includes just th
right amount of detail and clearly states his reasons for each step in exactly the way th
is advocated in this chapter. He also includes helpful explanations of where the proof
going, so that the reader has a better chance of understanding the big picture. In learnir
to write good proofs you can do no better than to study Euclid's proofs, especially tho:
from Book I of the Elements. As you come to master those proofs you will begin
appreciate them more and more. You will eventually find yourself reading and enjoyir
not just the proofs themselves, but also Heath's commentary [22] on the proofs. Hea
often explains why Euclid did things as he did and also indicates how other geomete
have proved the same theorem. Of course Euclid uses language quite differently fro
the way we do. His theorem statements themselves do not serve as good models of th
careful statements that modern standards of rigor demand.
CISES 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2F4fe13764-d811-43ff-a0ff-0cdba16270ce%2Fcyodouw_processed.png&w=3840&q=75)
Transcribed Image Text:1:44
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 wh
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 i
оn l.
Theorem 2.6.4. If P is any point, then there are at least two distinct lines l and m such th
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 a
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 i
оn l.
Theorem 2.6.7. There exist three distinct lines such that no point lies on all three of th
lines.
Theorem 2.6.8. If P is any point, then there exist points Q and R such that P, Q, and Ra
попcollinear.
Theorem 2.6.9. If P and Q are two points such that P + Q, then there exists a point R suc
that P, Q, and R are noncollinear.
Every theorem has a proper context and that context is an axiomatic system. Thi
every theorem has unstated hypotheses, namely that certain axioms are assumed true. F
example, the theorems above are theorems in incidence geometry. This means that eve:
one of them includes the unstated hypothesis that the three incidence axioms are assume
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
Euclid's proofs serve as excellent models for you to follow. Euclid usually includes just th
right amount of detail and clearly states his reasons for each step in exactly the way th
is advocated in this chapter. He also includes helpful explanations of where the proof
going, so that the reader has a better chance of understanding the big picture. In learnir
to write good proofs you can do no better than to study Euclid's proofs, especially tho:
from Book I of the Elements. As you come to master those proofs you will begin
appreciate them more and more. You will eventually find yourself reading and enjoyir
not just the proofs themselves, but also Heath's commentary [22] on the proofs. Hea
often explains why Euclid did things as he did and also indicates how other geomete
have proved the same theorem. Of course Euclid uses language quite differently fro
the way we do. His theorem statements themselves do not serve as good models of th
careful statements that modern standards of rigor demand.
CISES 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.
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