Question

Cho (a + b + c)² = 3(ab + bc + ca). Chứng minh rằng a = b = c

Giải thích rõ ràng nhé. 

Answer 1

(a+b+c)²=3(ab+bc+ca)

<=> a²+b²+c²+2ab+2ac+2bc=3ab+3bc+3ca

<=> a²+b²+c²+2ab+2ac+2bc-3ab-3bc-3ca=0

<=> a²+b²+c²-ab-bc-ca=0

<=> 2a²+2b²+2c²-2ab-2bc-2ca=0

<=> (a²-2ab+b²)+(b²-2bc+c²)+(c²-2ca+a²)=0

<=> (a-b)²+(b-c)²+(c-a)²=0

Vì \(\hept{\begin{cases}\left(a-b\right)^2\ge0\\\left(b-c\right)^2\ge0\\\left(c-a\right)^2\ge0\end{cases}}\)\(\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\)

Dấu "=" xảy ra khi \(\hept{\begin{cases}a-b=0\\b-c=0\\c-a=0\end{cases}\Rightarrow\hept{\begin{cases}a=b\\b=c\\c=a\end{cases}\Rightarrow}a=b=c}\) (đpcm)