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If a b c > 0 be real numbers such that n N, there exists triangles of side lengths a n , b n , c n . Then prove that those triangles must be Isosceles.

Answer & Explanation

Kiana Cantu

Kiana Cantu

Beginner2022-07-07Added 22 answers

Suppose the triangle is not isosceles. Then a > b > c > 0.
We need to show that for sufficiently large n, a n > b n + c n , which violates the triangle inequality.
Note that a n b n + c n a n 2 b n = 1 2 ( a b ) n as a b > 1.
Thus there exists some large N such that the quotient is larger than 1, and this completes the proof by contrapositive.

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