Answer: Option D
$$\eqalign{
& {\text{Given, }} \cr
& {a^2} + {b^2} + {c^2} - ab - bc - ca = 0 \cr
& {\text{Find, }}a:b:c = ? \cr
& {\text{According to the question, }} \cr
& {a^2} + {b^2} + {c^2} - ab - bc - ca = 0 \cr
& \Rightarrow {a^2} + {b^2} + {c^2} = ab + bc + ca \cr
& {\text{Let }}a = b = c = 1 \cr
& \Rightarrow {1^2} + {1^2} + {1^2} = 1 \times 1 + \left( {1 \times 1} \right) + \left( {1 \times 1} \right) \cr
& \Rightarrow 3 = 3 \cr
& \Rightarrow {\text{So, ratio of }}a:b:c = 1:1:1 \cr} $$