# Valid Inequality for a Triangle!

Geometry Level 5

$\large{\left( \dfrac{h_a^t + h_b^t + h_c^t}{3} \right)^{1/t} \leq \dfrac{\sqrt{3}}{2}\left( \dfrac{a^t + b^t + c^t}{3}\right)^{1/t}}$

Let $$a,b,c$$ be the sides of a triangle and $$h_a, h_b, h_c$$ respectively be the corresponding altitudes. If the maximum range of validity of the above inequality for $$t$$ be: $$\quad \large{-\alpha < t < \alpha}$$,

where $$t \neq 0$$ and $$\alpha = \dfrac{\ln(A)}{\ln(B)}$$, and where $$A,B \in \mathbb R$$, find the value of $$\dfrac{A}{B}$$.

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