Question

Chứng minh rằng:

a² + b² + c²=ab+bc+ca thì a=b=c

Answer 1

Ta có:

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

\(\Leftrightarrow2\left(a^2+b^2+c^2\right)=2\left(ab+bc+ac\right)\)

\(\Leftrightarrow2a^2+2b^2+2c^2=2ab+2bc+2ac\)

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

\(\Rightarrow\left[{}\begin{matrix}a-b=0\\a-c=0\\b-c=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}a=b\\a=c\\b=c\end{matrix}\right.\Rightarrow a=b=c\left(đpcm\right)\)