$VT=a^2(\frac{1}{2bc}+\frac{1}{c^2+ab}+\frac{1}{b^2+ac})+b^2(\frac{1}{2ac}+{\frac{1}{a^2+bc}+\frac{1}{c^2+ab}})+c^2(\frac{1}{2ab}+\frac{1}{a^2+bc}+\frac{1}{b^2+ac}) $$\geq \frac{9a^2}{b^2+c^2+2bc+ab+ac}+\frac{9b^2}{a^2+c^2+2ac+bc+ab}+\frac{9c^2}{a^2+b^2+2ab+bc+ac} $
$\geq \frac{9(a+b+c)^2}{2(a^2+b^2+c^2)+4(ab+bc+ca)}=\frac{9(a+b+c)^2}{2(a+b+c)^2}=\frac{9}{2}$