Question

cho a^2+b^2+c^2=ab+bc+ca . Chứng minh rằng a=b=c 

Answer 1

\(a^2+b^2+c^2=ab+ac+ca\)

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

\(2a^2+2b^2+2c^2-2ab-2bc-2ac=0\)

\(a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ac+a^2=0\)

\(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)

\(\left[\begin{array}{nghiempt}a-b=0\\b-c=0\\c-a=0\end{array}\right.\)

\(\left[\begin{array}{nghiempt}a=b\\b=c\\c=a\end{array}\right.\)

\(a=b=c\left(\text{đ}pcm\right)\)