A) \[a+b=c~\]
B) \[a+\text{ }c=b~~\]
C) \[b+c=a~~~\]
D) \[a=b=c\]
Correct Answer: D
Solution :
\[3\,\,({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] \[={{\left( a+b+c \right)}^{2}}\] \[=3{{a}^{2}}+3{{b}^{2}}+3{{c}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}\text{ }2ab+2bc+2ca\]\[2{{a}^{2}}+2{{b}^{2}}+2{{c}^{2}}-2ab-2bc-2ca=0\] \[\Rightarrow \] \[{{a}^{2}}+{{b}^{2}}-2ab+{{b}^{2}}+{{c}^{2}}-2bc+{{c}^{2}}+{{a}^{2}}-2ca=0\] \[\Rightarrow \]\[{{\left( a-b \right)}^{2}}+{{\left( b-c \right)}^{2}}+{{\left( c-a \right)}^{2}}=0\] \[\Rightarrow \]\[a-b=0:b-c=0;c-a=0\] \[\Rightarrow \] \[a\text{ }=b;\text{ }b=c\text{ };\text{ }c=a\] \[\therefore a=b=c\]
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