Date: Aug 22, 2011 8:45 AM Author: Avni Pllana Subject: Re: 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

> 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