Question

4)CMR nếu (a-b)²+(b-c)²+(c-a)²=(a+b-2c)²+(b+c-2a)²+(c+a-2b)² thì a=b=c

Answer 1

\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ac=\left(a+b\right)^2+\left(b+c\right)^2+\left(c+a\right)^2-4c\left(a+b\right)-4a\left(b+c\right)-4b\left(a+c\right)\)

\(\Leftrightarrow-4c\left(a+b\right)-4b\left(a+c\right)-4a\left(b+c\right)=2a^2+2b^2+2c^2-2ab-2bc-2ac-2a^2-2b^2-2c^2-2ac-2bc-2ab\)

\(\Leftrightarrow-4\left(ac+bc+ab+bc+ab+ac\right)=-4ab-4bc-4ac\)

\(\Leftrightarrow-4\left(2ab+2bc+2ac\right)+4\left(ab+bc+ac\right)=0\)

\(\Leftrightarrow-4\left(2ab+2bc+2ac-ab-ac-bc\right)=0\)

=>ab+bc+ac=0

=>a=b=c