[tex]\displaystyle \frac{x}{y+z}= \frac{y}{x+z}= \frac{z}{x+y} \\ \\ \\ \frac{x}{y+z}+1= \frac{y}{x+z}+1= \frac{z}{x+y}+1 \\ \\ \\ \frac{x+y+z}{y+z}= \frac{x+y+z}{x+z}= \frac{x+y+z}{x+y} \\ \\ \\ Deoarece~x,y,z \in Q_+ \Rightarrow x+y+z\ \textgreater \ 0 \Rightarrow y+z=x+z=x+y. \\ \\ Din~y+z=x+z ~rezulta~x=y.~~~~~~~~(1) \\ \\ Din~x+z=x+y ~rezulta~y=z.~~~~~~~~(2)[/tex]
[tex]\displaystyle Din~(1)~si~(2)~rezulta~\boxed{x=y=z}~. \\ \\ Cum~x=y~si~x=z,~rezulta~x+x=y+z \Leftrightarrow \boxed{2x=y+z}~.[/tex]
