a2 + b2 + c2 - ab - bc - ac = 0
⇔ a2 - 2ab + b2 + b2 - 2bc + c2 + c2 - 2ac + a2 = 0
⇔ ( a - b)2 + ( b - c)2 + ( c - a)2 = 0
⇔ a - b = 0 ; b - c = 0 và c - a = 0
⇔ a = b ; b = c và c = a
⇔ a = b = c
Đặt: A= \(a^2+b^2+c^2-ab-ac-bc\)=0
2A=\(2a^2+2b^2+2c^2-2ab-2ac-2bc=0\)
\(\Leftrightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(a^2-2ac+c^2\right)=0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)
Vì: \(\left(a-b\right)^2\ge0;\left(b-c\right)^2\ge0;\left(a-c\right)^2\ge0\forall a,b,c\)
\(\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\forall a,b,c\)
Dấu"=" xảy ra \(\Leftrightarrow a=b=c\)
