\(< =>2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)
\(< =>a^2-2ab+b^2+a^2-2ca+c^2+b^2-2bc+c^2=0\)
\(< =>\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\)
có \(\left\{{}\begin{matrix}\left(a-b\right)^2\ge0\\\left(a-c\right)^2\ge0\\\left(b-c\right)^2\ge0\end{matrix}\right.\) dấu"=" xảy ra<=>a=b=c
Ta có: \(a^2+b^2+c^2-ab-bc-ca=0\)
\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2ac-2bc=0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\)
Suy ra: a=b=c
<=>2a2+2b2+2c2−2ab−2bc−2ca=0<=>2a2+2b2+2c2−2ab−2bc−2ca=0
<=>a2−2ab+b2+a2−2ca+c2+b2−2bc+c2=0<=>a2−2ab+b2+a2−2ca+c2+b2−2bc+c2=0
<=>(a−b)2+(a−c)2+(b−c)2=0<=>(a−b)2+(a−c)2+(b−c)2=0
có ⎧⎪ ⎪⎨⎪ ⎪⎩(a−b)2≥0(a−c)2≥0(b−c)2≥0{(a−b)2≥0(a−c)2≥0(b−c)2≥0 dấu"=" xảy ra<=>a=b=c
=>2a2+2b2+2c2−2ab−2bc−2ca=0<=>2a2+2b2+2c2−2ab−2bc−2ca=0
<=>a2−2ab+b2+a2−2ca+c2+b2−2bc+c2=0<=>a2−2ab+b2+a2−2ca+c2+b2−2bc+c2=0
<=>(a−b)2+(a−c)2+(b−c)2=0<=>(a−b)2+(a−c)2+(b−c)2=0
có ⎧⎪ ⎪⎨⎪ ⎪⎩(a−b)2≥0(a−c)2≥0(b−c)2≥0{(a−b)2≥0(a−c)2≥0(b−c)2≥0 dấu"=" xảy ra<=>a=b=c
<=>2a2+2b2+2c2−2ab−2bc−2ca=0<=>2a2+2b2+2c2−2ab−2bc−2ca=0
<=>a2−2ab+b2+a2−2ca+c2+b2−2bc+c2=0<=>a2−2ab+b2+a2−2ca+c2+b2−2bc+c2=0
<=>(a−b)2+(a−c)2+(b−c)2=0<=>(a−b)2+(a−c)2+(b−c)2=0
có ⎧⎪ ⎪⎨⎪ ⎪⎩(a−b)2≥0(a−c)2≥0(b−c)2≥0{(a−b)2≥0(a−c)2≥0(b−c)2≥0 dấu"=" xảy ra<=>a=b=c