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7 3 I'm collecting proof's for Weitzenbock inequality, i made three proofs for this, See these proofs below(Whoever has a more cool proof please share). Let a, b, c be the sides of the triangle and A its area, prove that: $$a^{2}+b^{2}+c^{2}geq4sqrt{3}A$$ $$Proof 1$$ Let $R$ the circunradius.Suppose, by contradiction, that: begin{equation} frac{1}{a}+frac{1}{b}+frac{1}{c}<frac{3}{Rsqrt{3}} tag{1} end{equation} Using that: begin{equation} a+b+cleq 3Rsqrt{3} tag{2} end{equation} Multiplying (1) e (2): $$(a+b+c)left(frac{1}{a}+frac{1}{b}+frac{1}{c}right)<9$$ What is absurd, this can be seen using the inequality that relates the arithmetic mean to the harmonic mean, or by the inequality of Cauchy-Schwarz. From where we conclude: begin{equation} frac{1}{a}+frac{1}{b}+frac{1}{c}geqfrac{3}{R