\(a^2+b^2+c^2+ab+bc+ca=0\)
\(\Leftrightarrow2a^2+2b^2+2c^2+2ab+2bc+2ca=0\)
\(\Leftrightarrow a^2+2ab+b^2+b^2+2bc+c^2+c^2+2ca+a^2=0\)
\(\Leftrightarrow\left(a+b\right)^2+\left(b+c\right)^2+\left(c+a\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}a+b=0\\b+c=0\\c+a=0\end{cases}}\Leftrightarrow a=b=c=0\)