Without loss of generality let us assume that $a > b>c$
So $a^n < b^n + c^n$
Or $1 < (\frac{b}{a})^n + (\frac{c}{a})^n$
If $a>b > c$ then $(\frac{b}{a})^n = 0$ and $(\frac{c}{a})^n =0$ when n goes to infinite
So $1 < 0$ which is contradiction
So b has to be same as a for the sum to be 1 in the limit
Or the triangle is isosceles
Proved
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