Suppose a triangle has sides of length a, b and c.
Show that there exists a triangle with sides of length , and .

The lengths r, s and t form the sides of a triangle if and only if the
longest length is less than the sum of the other two lengths.

Suppose in our case that c is the longest length.
Let a = A^{2}, b = B^{2} and c = C^{2}. Then:

A^{2} + B^{2} > C^{2}

As a result:

(A + B)^{2} = A^{2} + B^{2} + 2AB > C^{2} + 2AB

So:

(A + B)^{2} > C^{2}

since 2AB is positive.
Hence

A + B > C

+ >

This establishes that

,

and

the lengths
of the sides of a triangle.