Question

Cho a^2+b^2+c^2=ab+bc+ca. Chứng minh rằng a=b=c.

Answer 1

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

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

\(\Rightarrow2.\left(a^2+b^2+c^2-ab-bc-ca\right)=2.0\)

\(\Rightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)

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

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

Mà \(\left(a-b\right)^2\ge0\)

\(\left(a-c\right)^2\ge0\)

\(\left(b-c\right)^2\ge0\)

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

\(\Rightarrow a=b=c\)

Vậy ...

Answer 2

thanks