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For which triangles with side length a, b and c do exist triangles with side length $a^n$, $b^n$ and $c^n$ for all positive integers $n\geq2$?

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The answer is

all isosceles triangles with sides a=b>=c.

All other triangles have

a longest side that will as the exponent n increases eventually exceed the sum of the two other sides.

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