Euclid's elements book 3 proposition 20

In summary, the theorem states that in a circle, the angle at the center is double the angle at the circumference when the angles have the same circumference as base. This can be proven by placing the two angles one on top of the other, but it is also valid if the angles have different portions of the circumference as long as they have the same length. However, in this case, the proposition may not hold if the corresponding arc is different. Therefore, it would be necessary to prove this as well.
  • #1
astrololo
200
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
 
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  • #2
Yes, that is true. As long as you have the same circumference cut off you have the same angles.
 
  • #3
Do you mean the situation like below?
http://imageshack.com/a/img540/6139/5K0JNE.png
 
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  • #4
If so, then it should be different, for the other angle is corresponding to the other arc.
 
  • #5
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 ?
 

Related to Euclid's elements book 3 proposition 20

1. What is Euclid's elements book 3 proposition 20?

Euclid's elements book 3 proposition 20 is a mathematical theorem that states that if two circles intersect, the line joining the centers is perpendicular to the line joining the points of intersection.

2. Why is Euclid's elements book 3 proposition 20 important?

This proposition is important because it is one of the fundamental theorems in Euclidean geometry and is used in many other proofs and constructions within the Elements.

3. Who was Euclid and why is he significant?

Euclid was a Greek mathematician who lived around 300 BC and is often referred to as the "Father of Geometry." He is significant because he wrote the Elements, a compilation of 13 books that laid the foundation for much of modern mathematics.

4. How does Euclid's elements book 3 proposition 20 relate to real-world applications?

While this theorem may not have direct real-world applications, it is a fundamental principle that is used in many geometric constructions and proofs. It also serves as the basis for more complex theorems and concepts in geometry.

5. Are there any other notable propositions in Euclid's elements book 3?

Yes, there are several other notable propositions in book 3, including proposition 1 which states that all angles inscribed in a semicircle are right angles, and proposition 31 which gives a construction for a regular pentagon. Book 3 also contains important theorems on the properties of triangles and circles.

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