Ta có : \(ax+by+cz=0\Rightarrow\left(ax+by+cz\right)^2=0\Rightarrow a^2x^2+b^2y^2+c^2z^2+2axby+2bycz+2czax=0\Rightarrow a^2x^2+b^2y^2+c^2z^2=-2abxy-2bycz-2czax\)
Xét tử số : \(bc\left(y-z\right)^2+ac\left(z-x\right)^2+ab\left(x-y\right)^2=bc\left(y^2-2yz+z^2\right)+ac\left(z^2-2xz+x^2\right)+ab\left(x^2-2xy+y^2\right)\)\(=bcy^2+bcz^2+acx^2+acz^2+abx^2+aby^2-2\left(bcyz+abxy+acxz\right)\)
\(=bcy^2+bcz^2+acx^2+acz^2+abx^2+aby^2+a^2x^2+b^2y^2+c^2z^2\)
\(=c\left(ax^2+by^2+cz^2\right)+b\left(ax^2+by^2+cz^2\right)+a\left(ax^2+by^2+cz^2\right)\)
\(=\left(a+b+c\right)\left(ax^2+by^2+cz^2\right)\)
\(\Rightarrow A=\frac{bc\left(y-z\right)^2+ac\left(z-x\right)^2+ab\left(x-y\right)^2}{ax^2+by^2+cz^2}=\frac{\left(a+b+c\right)\left(ax^2+by^2+cz^2\right)}{ax^2+by^2+cz^2}=a+b+c\)
