Date: Aug 16, 2011 4:25 AM
Author: ChessTal
Subject: Fundamental and trivial question on triangle inequality.

It's surprising (for me) that I will ask this but I have never met this problem before.

It's well known (e.g Recent Advances in Geometric Inequalities, Mitrinovic et al) that the following equivalence is true:

A,B,C are sides of a triangle if and only if A>0, B>0, C>0, A+B>C, A+C>B, C+B>A.

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:
If A>0, B>0, C>0, A+B>C, A+C>B, C+B>A then a triangle can be constructed with sides A, B, C.

**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.

For a better viewed version of this(using latex for better viewing of the equations and bold text) as also for some suggested solutions see here:

There, someone gave me an incomplete solution.
So as it seems, it suffices to show that the following implication is true:

A>0, B>0, C>0, A+B>C, A+C>B, B+C>A =>
(A^2 + B^2 + C^2)^2 > 2­·(A^4 + B^4 + C^4)

But i can't really seem to show that also. :(

But what impresses me most, is the lack of any reference i'm noticing(i have posted this in 3 big math forums and i got zero replies about any references), of a book or paper about this kind of fundamental theorem and a proof of it. Such an elementary and important theorem and not being included in geometry books is very bizarre fact for me!?!
Moreover the lack of a (strict and not descriptive of course) geometric proof of it is also odd for me.

Thanks in advance.