Lời giải:
Từ \(\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=2\)
\(\Rightarrow (x+y+z)\left(\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}\right)=2(x+y+z)\)
\(\Leftrightarrow \frac{x^2}{y+z}+\frac{xy}{x+z}+\frac{xz}{x+y}+\frac{xy}{y+z}+\frac{y^2}{x+z}+\frac{zy}{x+y}+\frac{xz}{y+z}+\frac{zy}{x+z}+\frac{z^2}{x+y}=2(x+y+z)\)
\(\Leftrightarrow \frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y}+\frac{xy+zy}{x+z}+\frac{xz+yz}{x+y}+\frac{xy+xz}{y+z}=2(x+y+z)\)
\(\Leftrightarrow \frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y}+y+z+x=2(x+y+z)\)
\(\Leftrightarrow \frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y}=x+y+z\) (đpcm)