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:

A,B,C are sides of a triangleif and only ifA>0, B>0, C>0, A+B>C, A+C>B, C+B>A

Of course the $\displaystyle \Rightarrow$ 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 $\displaystyle \Leftarrow$ 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>Athena 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.

Thanks in advance.