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August 15th, 2011, 03:26 AM  #1 
Newbie Joined: Jan 2010 Posts: 8 Thanks: 0  Fundamental and trivial question on triangle inequality.
It's surprising (for me) that I will ask this but I have never met this. It's well known (e.g Recent Advances in Geometric Inequalities, Mitrinovic et al) that the following is true: [color=#0000BF]A,B,C are sides of a triangle[/color] if and only if [color=#0000BF]A>0, B>0, C>0, A+B>C, A+C>B, C+B>A[/color] Of course the part of the above equivalence is well known and it has many proofs and also a simple geometric one that Euclid gave ........ all these are well known. You will find this implication in all books of geometry in the initial chapters, as also being followed with the simple proof I've mentioned. But what about the part of the equivalence? I have never seen a proof for this. Can anyone provide one, as also a reference for it(a book or paper for example)? As crazy as it looks, but looking the half internet didn't result in anything! So to be clear I'm speaking about proving the following theorem as also a reference for the proof: [color=#0000BF]If A>0, B>0, C>0, A+B>C, A+C>B, C+B>A[/color] then [color=#0000BF]a triangle can be constructed with sides A, B, C.[/color] **By saying "constructed" above, I don't obviously mean with compass and ruler construction, but I'm referring to the existence of a triangle with sides A, B, C. Thanks in advance. 
August 15th, 2011, 04:12 AM  #2 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: Fundamental and trivial question on triangle inequality.
The triangle with vertices at (0,0), (A,0) and (x,y) where has sides with length A, B and C. If you can show that the conditions on side length imply that is real, then you're done  I'm sure this isn't too tricky, but I'll leave it as an exercise (i.e. I couldn't be bothered right now ) 
August 15th, 2011, 04:23 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,972 Thanks: 2222 
Using the letters a, b and c for the sides, where a ? b ? c, draw a side of length c, then construct sides with lengths a and b in the usual way. It's easy to show the construction must succeed in producing a triangle if a + b > c.

August 15th, 2011, 04:37 AM  #4  
Newbie Joined: Jan 2010 Posts: 8 Thanks: 0  Re: Quote:
Quote:
That would make more sense, but was that really a typo?  
August 15th, 2011, 05:33 AM  #5 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: Fundamental and trivial question on triangle inequality.
No, it's correct as originally stated. Note that if we multiply A, B and C by a single positive parameter then the values of x and y should also increase by this parameter. Note that so the formula is correct in terms of dimensionality. 
August 15th, 2011, 09:46 AM  #6  
Newbie Joined: Jan 2010 Posts: 8 Thanks: 0  Re: Fundamental and trivial question on triangle inequality. Quote:
So all in all what we have to prove is the validity of the following implication: I can't really seem to manage solving this but i will try a bit more. But what is more important to me right now is the lack of any reference i'm noticing, of a book or paper about this kind of fundamental theorem and about a proof of it. Such an elementary and important theorem and not being included in geometry books is very bizarre fact for me. And even worse, even if i see the solution of the above implication i gave, i would still not be 100% satisfied since i will still miss a geometric proof. Or there isn't any? This can't be the case....  
August 15th, 2011, 05:18 PM  #7  
Global Moderator Joined: Dec 2006 Posts: 20,972 Thanks: 2222  Quote:
 

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fundamental, inequality, question, triangle, trivial 
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