$Iftherootsoftheequation(x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0areequalthen$

$\begin{array}{l}GivenEquation{x}^2-(b+c)x+bc+x^2-(a+c)x+ac+x^2-(a+b)x+ab=0\\ \Rightarrow 3x^2-2(a+b+c)x+(ab+bc+ca)=0\\ RootsareEqual\Rightarrow D=0\\ \Rightarrow 4{(a+b+c)}^2-4\left(3\right)(ab+bc+ca)=0\\ \Rightarrow {(a+b+c)}^2-3(ab+bc+ca)=0\\ \Rightarrow {(a-b)}^2+{(b-c)}^2+{(c-a)}^2=0\\ \Rightarrow a=b=c\\ \Rightarrow a+bw+c{w}^2=a(1+w+w^2)=a\left(0\right)\Rightarrow 0\end{array}$