Áp dụng BĐT AM-GM ta có:
\(x+y\ge2\sqrt{xy}\)
\(y+z\ge2\sqrt{yz}\)
\(x+z\ge2\sqrt{xz}\)
Cộng theo vế các BĐT trên ta có:
\(x+y+y+z+z+x\ge2\sqrt{xy}+2\sqrt{yz}+2\sqrt{xz}\)
\(\Leftrightarrow2x+2y+2z\ge2\sqrt{xy}+2\sqrt{yz}+2\sqrt{xz}\)
\(\Leftrightarrow2\left(x+y+z\right)\ge2\left(\sqrt{xy}+\sqrt{yz}+\sqrt{xz}\right)\)
\(\Leftrightarrow x+y+z\ge\sqrt{xy}+\sqrt{yz}+\sqrt{xz}\)
Dấu "=" xảy ra khi \(x=y=z\)
Nhân 2 vế đẳng thức với 2 ta được:
\(x+y+z=\sqrt{xy}+\sqrt{yz}+\sqrt{xz}\)
\(\Leftrightarrow2\left(x+y+z\right)=2\left(\sqrt{xy}+\sqrt{yz}+\sqrt{xz}\right)\)
\(\Leftrightarrow\left(\sqrt{x}-\sqrt{y}\right)^2+\left(\sqrt{y}-\sqrt{z}\right)^2+\left(\sqrt{z}-\sqrt{x}\right)^2=0\)
\(\Leftrightarrow x=y=z\) (Đpcm)
