Euclid's elements book 3 proposition 20

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Discussion Overview

The discussion revolves around a theorem related to angles in a circle, specifically Euclid's Elements Book 3 Proposition 20, which states that the angle at the center of a circle is double the angle at the circumference when the angles subtend the same arc. Participants explore the implications of this theorem and whether the angles must be positioned directly on top of one another or can be separated while still maintaining the same arc length.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant proposes that the theorem holds true even if the angles are not positioned directly on top of each other, as long as they subtend arcs of the same length.
  • Another participant agrees with this assertion, stating that having the same arc length results in the same angles.
  • A different participant questions whether the angles can be separated and still yield the same result, suggesting that the angles correspond to different arcs if they are not aligned.
  • A later reply reiterates the agreement that the angles can be proven to be equal if they subtend the same arc, indicating a need for further proof.

Areas of Agreement / Disagreement

There is some agreement among participants that angles subtending the same arc are equal, but there is disagreement regarding whether the angles must be aligned or can be separated. The discussion remains unresolved on this point.

Contextual Notes

The discussion does not clarify the assumptions regarding the positioning of angles or the implications of different arc lengths, leaving these aspects open to interpretation.

astrololo
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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
 
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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 ?
 

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