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

  • Thread starter astrololo
  • Start date
  • #1
200
3

Main Question or Discussion Point

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!
 

Answers and Replies

  • #2
FactChecker
Science Advisor
Gold Member
5,580
2,057
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
Merlin3189
Homework Helper
Gold Member
1,576
717
... 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
FactChecker
Science Advisor
Gold Member
5,580
2,057
I'm just wondering what you mean by "diagonal" here?
Oh, I meant diameter. Thanks for the correction.
 
  • #5
Merlin3189
Homework Helper
Gold Member
1,576
717
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
FactChecker
Science Advisor
Gold Member
5,580
2,057
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
200
3
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 Threads on On the same straight line there cannot be constructed two...

Replies
2
Views
769
Replies
3
Views
3K
  • Last Post
2
Replies
35
Views
7K
  • Last Post
Replies
7
Views
6K
  • Last Post
Replies
13
Views
876
  • Last Post
Replies
3
Views
8K
  • Last Post
Replies
4
Views
8K
  • Last Post
Replies
11
Views
885
Replies
1
Views
1K
  • Last Post
Replies
6
Views
6K
Top