Theorem : Let be three real numbers. Then they are the sides of a triangle if and only if there are positive real numbers such that .
For the first direction : If are positive real numbers then clearly form side lengths of a triangle.
For the other direction : If form side lengths of a triangle then we put . Then easily we see that, and .
This completes the proof of the theorem .
Another way of proving (this is more natural and beautiful) the latter part is as follows: Let have side lengths . Now consider the incircle (a.k.a. the inscribed circle) of . Let be the points on the sides of where the incircle touches the sides of the triangle. Then using the fact that tangents drawn from an external point to a circle are equal in length gives the desired result.
This simple theorem can be used to solve many problems (especially from mathematical inequalities and geometry).
For example you can try to use it to solve the following problem :
If are side lengths of a triangle then show that must also be side lengths of a triangle.