<s>[INST] Inequality between altitude and sides in triangle Let $a,b,c$ be the side lengths and $h_a,h_b,h_c$ the altitudes each connect a vertex to the opposite side and are perpendicular to that side. Then we need to prove $h_a^2+h_b^2+h_c^2\leq\dfrac14(a+b+c)^2$. 
I know the inequality $h_a^2+h_b^2+h_c^2\leq\dfrac34(a^2+b^2+c^2)$ by using Cauchy inequality. But I could not extend the proof for the above inequality. Does one help me to prove this? 
 [/INST] Following ASCII advocate's idea, the problem boils down to proving that:
$$ 4\Delta^2\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\leq (a+b+c)^2 \tag{1}$$
or, using Heron's formula,
$$ (a+b-c)(a-b+c)(-a+b+c)\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\leq (a+b+c). \tag{2}$$
Through Ravi substitution that turns out to be equivalent to:
$$ 8xyz\left(\frac{1}{(x+y)^2}+\frac{1}{(x+z)^2}+\frac{1}{(y+z)^2}\right)\leq 2(x+y+z)\tag{3} $$
or to:
$$ \sum_{cyc}\frac{1}{(x+y)^2}\leq\sum_{cyc}\frac{1}{4xy}\tag{4}$$
that is trivial as a consequence of the AM-GM inequality.
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