Condition on sides of triangle to proof it is isoceles
If the triangle with sides $a,b,c$ were not isosceles, then $a>b>c>0$. Then, for $n$ sufficiently big, $a^n>b^n+c^n$, against the assumption that $a^n,b^n$ and $c^n$ are the lengths of the sides of a triangle. It means that at least one equality of $a\geq b\geq c$ must hold (actually, we can even say that $a=b$).