Question

CMR nếu a² +b² + c​2=ab+ac+bc thì a=b=c

Answer 1

Từ \(a^2+b^2+c^2=ab+bc+ac\Rightarrow a^2+b^2+c^2-ab-bc-ac=0\)

\(\Rightarrow2\left(a^2+b^2+c^2-ab-bc-ac\right)=0\)

\(\Rightarrow2a^2+2b^2+2c^2-2ab-2bc-2ac=0\)\(\Rightarrow\left(a^2-2ab+b^2\right)+\left(a^2-2ac+c^2\right)+\left(b^2-2bc+c^2\right)=0\)

\(=\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\)

\(\Rightarrow\hept{\begin{cases}\left(a-b\right)^2=0\\\left(a-c\right)^2=0\\\left(b-c\right)^2=0\end{cases}}\Rightarrow\hept{\begin{cases}a-b=0\\a-c=0\\b-c=0\end{cases}\Rightarrow a=b=c}\)

Vậy nếu \(a^2+b^2+c^2=ab+bc+ac\)thì \(a=b=c\)