
Re: Fundamental and trivial question on triangle inequality.
Posted:
Aug 22, 2011 8:45 AM


> 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 > f 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 > 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 > e 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 > r better viewing of the equations and bold text) as > also for some suggested solutions see here: > http://www.mymathforum.com/viewtopic.php?f=13&t=22545& > start=0 > > There, someone gave me an incomplete solution. > So as it seems, it suffices to show that the > e 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.
We start from
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) ...(1)
If we rearrange the last inequality, we obtain
A^4 + B^4 + C^4 < 2*A^2*B^2 + 2*B^2*C^2 + 2*C^2*A^2 ...(2)
From the Law of Cosines we have
A^2 = B^2 + C^2 2*B*C*cos(A) , or
A^2  B^2  C^2 = 2*B*C*cos(A) ... (3)
Squareing both sides of (3), we have
(A^2  B^2  C^2)^2 = 4*B^2*C^2*(cos(A))^2 , or
(A^2  B^2  C^2)^2 < 4*B^2*C^2 ,...(4)
since (cos(A))^2 < 1.
Finally from (4) follows (2) and (1).
Best regards, Avni

