Question

chứng minh rằng nếu\(\sqrt{x}+\sqrt{y}-\sqrt{z}=0\)    thì \(\frac{1}{y+z-x}+\frac{1}{z+x-y}+\frac{1}{x+y-z}=0\)

Answer 1

\(\sqrt{z}=\sqrt{x}+\sqrt{y}\Rightarrow z=x+y+2\sqrt{xy}\Rightarrow x+y-z=-2\sqrt{xy}\)

\(\sqrt{y}=\sqrt{z}-\sqrt{x}\Rightarrow y=x+z-2\sqrt{zx}\Rightarrow z+x-y=2\sqrt{zx}\)

\(\sqrt{x}=\sqrt{z}-\sqrt{y}\Rightarrow x=y+z-2\sqrt{yz}\Rightarrow y+z-x=2\sqrt{yz}\)

\(\frac{1}{y+z-x}+\frac{1}{z+x-y}+\frac{1}{x+y-z}=\frac{1}{2}\left(\frac{1}{\sqrt{zx}}+\frac{1}{\sqrt{yz}}-\frac{1}{\sqrt{xy}}\right)\)

\(=\frac{1}{2}.\frac{\sqrt{x}+\sqrt{y}-\sqrt{z}}{\sqrt{xyz}}=0\)