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b, Ta có \(m=a+b+c\)
\(\Rightarrow am+bc=a\left(a+b+c\right)+bc=a\left(a+b\right)+ac+bc=\left(a+c\right)\left(a+b\right)\)
CMTT \(bm+ac=\left(b+c\right)\left(b+a\right)\);\(cm+ab=\left(c+a\right)\left(c+b\right)\)
Suy ra \(\left(am+bc\right)\left(bm+ac\right)\left(cm+ab\right)=\left(a+b\right)^2\left(a+c\right)^2\left(b+c\right)^2\)
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Ta có: \(a^2+b^2+c^2=ab+bc+ac\)
\(\Rightarrow2.\left(a^2+b^2+c^2\right)=2.\left(ab+bc+ac\right)\)
\(\Rightarrow2a^2+2b^2+2c^2=2ab+2bc+2ac\)
\(\Rightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2ac+c^2\right)+\left(c^2-2ac+a^2\right)\)
\(\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(a-c\right)^2\ge0\) (BĐT luôn đúng)
Dấu "=" xảy ra khi và chỉ khi \(\hept{\begin{cases}\left(a-b\right)^2=0\\\left(b-c\right)^2=0\\\left(a-c\right)^2=0\end{cases}}\Leftrightarrow\hept{\begin{cases}a=b\\b=c\\c=a\end{cases}\Leftrightarrow a=b=c}\)
\(a^2+b^2+c^2=ab+bc+ac\)
\(\Leftrightarrow2\left(a^2+b^2+c^2\right)=2\left(ab+bc+ac\right)\)
\(\Leftrightarrow2a^2+2b^2+2c^2=2ab+2bc+2ac\)
\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ac=0\)
\(\Leftrightarrow\left(a^2-2ac+c^2\right)+\left(b^2-2ab+a^2\right)+\left(c^2-2bc+b^2\right)=0\)
\(\Leftrightarrow\left(a-c\right)^2+\left(b-a\right)^2+\left(c-b\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}a-c=0\\b-a=0\end{cases}}\) \(\Leftrightarrow a=b=c\left(đpcm\right)\)
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\(a^2+b^2+c^2=ab+bc+ca\)
\(\Leftrightarrow2a^2+2b^2+2c^2=2ab+2bc+2ca\)
\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)
\(\Leftrightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(a^2-2ca+c^2\right)=0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(a-c\right)^2=0\)
Mà \(\left(a-b\right)^2+\left(b-c\right)^2+\left(a-c\right)^2\ge0\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(a-b\right)^2=0\\\left(b-c\right)^2=0\\\left(a-c\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=b\\b=c\\c=a\end{matrix}\right.\Leftrightarrow a=b=c\)
\(\Rightarrowđpcm\)
Vậy...
Ta có: a2 + b2 + c2 = ab + bc + ca
=> a2 + b2 + c2 - ab - bc - ca = 0
=> (a-b)2 + (b-c)2 + (c-a)2 = 0
=> đpcm.