Question

a^{2 +} b^{2 +} c^{2 = AB + BC +CA} 

CMR A=B=C

Answer 1

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

\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2ac-2bc=0\)

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

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

\(\Leftrightarrow\left\{{}\begin{matrix}a-c=0\\a-b=0\\b-c=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=c\\a=b\\b=c\end{matrix}\right.\)\(\Rightarrow a=b=c\)