Euclid's elements book 3 proposition 20

  • Thread starter Thread starter astrololo
  • Start date Start date
  • Tags Tags
    Book Elements
AI Thread Summary
The theorem states that in a circle, the angle at the center is double the angle at the circumference when both angles share the same arc as a base. The discussion confirms that the angles can indeed be positioned differently along the circumference, as long as they subtend arcs of equal length, maintaining the validity of the proposition. However, it is noted that if the angles correspond to different arcs, the relationship may not hold, requiring further proof. The conversation emphasizes the importance of understanding the geometric relationships involved in the theorem. Clarity in these concepts is essential for accurate proof verification in Euclidean geometry.
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 ?
 
Thread 'Video on imaginary numbers and some queries'

Thread 'Video on imaginary numbers and some queries'

Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »
Thread 'Unit Circle Double Angle Derivations'

Thread 'Unit Circle Double Angle Derivations'

Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
View full post »

Similar threads

Horn angles and Euclid's elements
Replies
2
Views
2K
Why do we need postulate 4 in Euclid's element (P14)?
Replies
2
Views
2K
Is Euclid's Fourth Postulate Redundant?
Replies
3
Views
2K
On the same straight line there cannot be constructed two....
Replies
6
Views
1K
Euclid's elements proposition 15 book 3
Replies
2
Views
1K
To circumscribe a square about a given circle
Replies
2
Views
1K
A How Does Non-Euclidean Geometry Differ from Euclid's Postulates?
Replies
3
Views
2K
Triangles which are on the same base and in the same paralle
Replies
6
Views
1K
Suggestions for HS geometry book (proofs)
Replies
14
Views
1K
Euclid's Elements: Proposition 7
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top