On the same straight line there cannot be constructed two....

In summary, the conversation discusses the possibility of building similar segments of circles, with equal angles, but unequal ones. The definition of similar segments of circles is also mentioned. The conversation also explores the concept of using a diameter as the base for two similar segments, and whether they would be equal or not. Ultimately, it is concluded that similar segments of circles can exist on both sides of a line, but they must have equal bases in order to be truly equal.
  • #1
astrololo
200
3
So, according to this figure :http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII23.html We cannot have similar segments of circles and unequal ones be built on the same side of the same straight line.

My question is : Can we build similar segments of circles but unequal ones ? (It seems to imply it)

The definition of similar segments of circles is : "Similar segments of circles are those which admit equal angles, or in which the angles equal one another."

Here is the link for it : http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/defIII11.html

Thank you!
 
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  • #2
Are you asking if it is possible when you omit "same side"? I would say yes, because any diagonal has two similar, unequal segments on opposite sides.
 
  • #3
FactChecker said:
... I would say yes, because any diagonal has two similar, unequal segments on opposite sides.
I'm just wondering what you mean by "diagonal" here? Is it the same as "chord" ?
If you are referring to the segments on the opposite side of a chord, they do not seem to be similar as the subtended angles are supplementary rather than equal.
 
  • #4
Merlin3189 said:
I'm just wondering what you mean by "diagonal" here?
Oh, I meant diameter. Thanks for the correction.
 
  • #5
Well, if it's a diameter, then the two parts would be identical, so similar but not unequal. Putting them on opposite sides doesn't help, they are still equal.

Going back to the OP I share his bewilderment: I can't see why which side of the line they are makes any difference. Putting unequal segments on opposite sides would never make them similar. As far as I can see, the only reason for drawing them on the same side is to make the diagram for his proof work.
But to me it is a strange proof, using a non-obvious fact (angles in a segment are equal) to prove an obvious one!

As far as the question, "Can we build similar segments of circles but unequal ones ?" goes, if the line AB means the line starting at A and ending at B, and this has to be the chord of the circle, then no.
To me, "the line AB" has always meant "the line which passes through A and B, extended infinitely in both directions". In that case we could have any number of similar segments (enlargements of each other) which need not be equal sitting on that line on either side.
Just as similar triangles (or any other shapes) with equal bases are equal, then similar segments with equal chords are equal. So you could not have unequal similar shapes with the same base, whichever side of a line they were on.
 
  • #6
Merlin3189 said:
Well, if it's a diameter, then the two parts would be identical, so similar but not unequal.
Are they talking about equal length or equal? The two are similar and equal length, but not equal (they are not the same segment). I think that is the only interpretation of the proposition that would make it true.
 
  • #7
By the way, the base of the segment doesn't need to be equal. My question would be, can we have two unequal segments of circles that are similar, but they don't need to have equal bases.
 

Related to On the same straight line there cannot be constructed two....

What does "On the same straight line there cannot be constructed two..." mean?

This statement is known as the "parallel postulate" and it means that two lines on a plane will never intersect, no matter how far they are extended.

Why is this statement important in mathematics?

The parallel postulate is important because it is one of the five postulates in Euclidean geometry, which forms the basis for many mathematical proofs and constructions.

Is there any proof for this statement?

Yes, there are multiple proofs for the parallel postulate, including the famous "Playfair's axiom" which states that given a line and a point not on the line, there is only one line parallel to the given line through the given point.

Can this statement be applied to three-dimensional space?

No, the parallel postulate only applies to two-dimensional planes. In three-dimensional space, there can be multiple lines that are parallel to each other and do not intersect.

Are there any exceptions to this statement?

Yes, there are non-Euclidean geometries, such as hyperbolic geometry, where the parallel postulate does not hold true. In these geometries, there can be multiple lines that are parallel to a given line and do intersect at some point.

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