Euclid's elements book 3 proposition 20

  • Context: Undergrad 
  • Thread starter Thread starter astrololo
  • Start date Start date
  • Tags Tags
    Book Elements
Click For Summary
SUMMARY

The discussion centers on Euclid's Elements Book 3 Proposition 20, which states that the angle at the center of a circle is double that at the circumference when both angles subtend the same arc. Participants confirm that the angles can indeed be positioned differently along the circumference, provided they subtend arcs of equal length. The proof can be simplified by aligning the angles, but this is not a strict requirement for the theorem to hold true. Further verification of this understanding is suggested for clarity.

PREREQUISITES
  • Understanding of basic circle geometry
  • Familiarity with Euclidean geometry principles
  • Knowledge of angle relationships in circles
  • Ability to interpret geometric proofs
NEXT STEPS
  • Study the proof of Euclid's Elements Book 3 Proposition 20 in detail
  • Explore the concept of angles subtended by arcs in circle geometry
  • Learn about geometric proof techniques and their applications
  • Investigate related propositions in Euclid's Elements for deeper insights
USEFUL FOR

Students of geometry, educators teaching Euclidean principles, and anyone interested in the foundations of geometric proofs will benefit from this discussion.

astrololo
Messages
200
Reaction score
3
I have the following theorem : "In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base."

(Figure is in the link) http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII20.html

English isn't my first language, so I just want to make sure that I understood something correctly. We prove the theorem by putting the two angles one on the other for the circumference. I was just wondering, can I assume that the angles do not need to be one on the other and they can have different portion of the circumference, as long as the circumference are of the same length ? (Will the proposition still work in this way?) I guess that Euclid did the proof by putting the angles one on the other for making the demonstration less wordy. (Less long to read)

Thank you!

geometry proof-verification euclidean-geometry
 
Mathematics news on Phys.org
Yes, that is true. As long as you have the same circumference cut off you have the same angles.
 
Do you mean the situation like below?
http://imageshack.com/a/img540/6139/5K0JNE.png
 
Last edited by a moderator:
If so, then it should be different, for the other angle is corresponding to the other arc.
 
HallsofIvy said:
Yes, that is true. As long as you have the same circumference cut off you have the same angles.
I guess that I would also need to prove this then. right ?
 

Similar threads

Graduate Horn angles and Euclid's elements
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Why do we need postulate 4 in Euclid's element (P14)?
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Euclid's elements proposition 15 book 3
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad On the same straight line there cannot be constructed two....
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad To circumscribe a square about a given circle
  • · Replies 2 ·
Replies
2
Views
1K
Graduate How Does Non-Euclidean Geometry Differ from Euclid's Postulates?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Triangles which are on the same base and in the same paralle
  • · Replies 6 ·
Replies
6
Views
1K
Undergrad How did Euclid go about forming his propositions?
  • · Replies 13 ·
Replies
13
Views
3K
Undergrad Compilations of proofs of Euclid's Theorem on primes
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Euclid's Elements: Proposition 7
  • · Replies 1 ·
Replies
1
Views
2K