Question

cho\(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=4.\left(a^2+b^2+c^2-ab-ac-bc\right)\)

chứng minh:a=b=c

Answer 1

Có \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=4\left(a^2+b^2+c^2-ab-ac-bc\right)\)

\(\Rightarrow a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ac+a^2-4a^2-4b^2-4c^2+4ab+4ac+4bc=0\)

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

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

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

\(\Rightarrow\left[{}\begin{matrix}\left(a-b\right)^2=0\\\left(b-c\right)^2=0\\\left(a-c\right)^2=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}a-b=0\\b-c=0\\a-c=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}a=b\\b=c\\a=c\end{matrix}\right.\Rightarrow a=b=c\left(đpcm\right)\)