If $y=-x$ and $\displaystyle \frac{y}{x-z}=\frac{x}{y}$ then either $x:y:z=1:-1:0$ or $x:y:z=-1:+1:0$.
Is this correct? If not why?
If $y=-x$ and $\displaystyle \frac{y}{x-z}=\frac{x}{y}$ then either $x:y:z=1:-1:0$ or $x:y:z=-1:+1:0$.
Is this correct? If not why?
If $z=x$ then $x-z = 0$ and then the LHS is undefined.
Since $x=-y$ the RHS is equal to $-1$, therefore $\frac{y}{x-z} = -1$, which means $y = z-x$ and so $z=0$.
Note, however, that $x,y$ can be pretty much anything under $z=0$, not just $\pm 1$.
So long x $\neq$ z and y$\neq$ 0. This should be stated along with the derived conclusion about the ratios.