Question

Chứng minh rằng:
a² + b² + c² = ab + ac + bc => a = b = c

Answer 1

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

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

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

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

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

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

Answer 2

Answer 3