\(a+b+c=0\Rightarrow\left\{{}\begin{matrix}a+b=-c\\a+c=-b\\b+c=-a\end{matrix}\right.\)
\(\Rightarrow VT=\dfrac{ab}{a^2+b^2-c^{^2}}+\dfrac{bc}{b^2+c^2-a^{^2}}+\dfrac{ca}{c^2+a^2-b^{^2}}\\ =\dfrac{ab}{a^2+\left(b+c\right)\left(b-c\right)}+\dfrac{bc}{b^2+\left(c+a\right)\left(c-a\right)}+\dfrac{ca}{c^2+\left(a+b\right)\left(a-b\right)}\\ =\dfrac{ab}{a^2-a\left(b-c\right)}+\dfrac{bc}{b^2-b\left(c-a\right)}+\dfrac{ca}{c^2-c\left(a-b\right)}\\ =\dfrac{b}{a-b+c}+\dfrac{c}{b-c+a}+\dfrac{a}{c-a+b}\\ =\dfrac{b}{\left(a+c\right)-b}+\dfrac{c}{\left(a+b\right)-c}+\dfrac{a}{\left(c+b\right)-a}\\ =\dfrac{b}{-b-b}+\dfrac{c}{-c-c}+\dfrac{a}{-a-a}\\ =\dfrac{b}{-2b}+\dfrac{c}{-2c}+\dfrac{a}{-2a}\\ =-\dfrac{1}{2}-\dfrac{1}{2}-\dfrac{1}{2}=-\dfrac{3}{2}=VP\)
