TITLE: Isolating variables in three-variable equation system
QUESTION [5 upvotes]: I am presented with the following equation system
$x^2 = 8+(y-z)^2$
$y^2 = 12+(z-x)^2$
$z^2 = 24+(x-y)^2$
I don't even know where to start. I am assuming you need to isolate the variables to begin with but even then - I have been unable to reach any sort of progress.
Please help. Thank you.

REPLY [3 votes]: It's $$(x-y+z)(x-z+y)=8$$
$$(y-x+z)(y-z+x)=12$$ and
$$(z-x+y)(z-y+x)=24,$$ which gives
$$\prod_{cyc}(x+y-z)^2=48^2.$$
Now, if $\prod\limits_{cyc}(x+y-z)=48$ we obtain:
$$y+z-x=6,$$
$$x+z-y=4$$ and
$$x+y-z=2,$$ which gives
$$(x,y,z)=(3,4,5).$$
The second case gives $$(x,y,z)=(-3,-4,-5).$$