Suppose $x,y,z\neq 0$, that $(x,y,z)$ is a point in $\mathbb{R}^3$ and that $td[\mathbb{Q}(x,y,z):\mathbb{Q}]=2$ (where $td[,]$ denotes the transcendence degree of the field extension).
Is it true that the set $\{x/z,y/z\}$ is algebraically independent over $\mathbb{Q}$?
Edit: I'm really interested in the case when $(x,y,z)$ is a point on the unit sphere (about the origin). In trying to make the question more simple, I've probably missed several special cases where the answer to the question as stated is obvious. One such example is given by Wojowu's comment below.
To avoid all special cases, probably what I want is to assume that $td[\mathbb{Q}(x,y):\mathbb{Q}]=2$, $td[\mathbb{Q}(x,z):\mathbb{Q}]=2$ and $td[\mathbb{Q}(y,z):\mathbb{Q}]=2$.
No: take for example $x=\alpha^2$, $y=\beta^2$, and $z=\alpha\beta$, where $\alpha$ and $\beta$ are transcendental numbers that are algebraically independent.