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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Question 8
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2Fa6fc492c-a7e0-42cc-8ee0-67f2996d3baa%2Fe3c5jko_processed.png&w=3840&q=75)
Transcribed Image Text: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
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