ABC26GN3193 · Ratio and Proportion

$a^{2}+b^{2}+c^{2}-a b-b c-c a=0$ Multiple equation (i) by 2 we get $$\begin{aligned} & \Rightarrow 2 a^{2}+2 b^{2}+2 c^{2}-2 a b-2 b c-2 c a=0 \\ & \Rightarrow\left(a^{2}+b^{2}-2 a b\right)+\left(b^{2}+c^{2}-2 b c\right)+\left(c^{2}+a^{2}-2 c a\right)=0 \end{aligned} \Rightarrow(a-b)^{2}+(b-c)^{2}+(c-a)^{2}=0$$ [If $x^{2}+y^{2}+z^{2}=0$ then $x=0, y=0, z=0$ ] $$\begin{aligned} & \therefore a-b=0 \Rightarrow a=b \\ & b-c=0 \Rightarrow b=c \\ & c-a=0 \Rightarrow c=a \\ & \therefore a=b=c \\ & \therefore a: b: c=1: 1: 1 \end{aligned}$$