$$\begin{gathered} Therootsoftheequation(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 \\ arereal andequalif \end{gathered}$$

$\begin{array}{l}GivenEquation(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0\\ \Rightarrow 3x^2-2x(a+b+c)+(ab+bc+ca)=0\\ Discri\min ant=0\\ \Rightarrow 4{(a+b+c)}^2-12(ab+bc+ca)=0\\ \Rightarrow 4[{(a+b+c)}^2-3(ab+bc+ca)]=0\\ \Rightarrow 2[{(a-b)}^2+{(b-c)}^2+{(c-a)}^2]=0\\ \Rightarrow {(a-b)}^2+{(b-c)}^2+{(c-a)}^2=0\\ \Rightarrow a-b=0,b-c=0,c-a=0\\ \Rightarrow a=b=c\end{array}$