Solving the system $a^2-c^2=x^2-z^2$, $ab=xy$, $ac=xz$, $bc=yz$ I've stumbled upon these equations, and am struggling to find a manual way to solve this in $\Re$:

$$\begin{align}
a^2-c^2&=x^2-z^2 \\
ab&=xy \\
ac&=xz \\
bc&=yz\end{align}$$

I've used Wolfram Alpha to compute this and I found that this is only possible if:
$$a=x, \quad y=b, \quad z=c$$
Is there an easy manual way to compute this?
I've tried moving around variables with no luck.

This is an homogeneous system so making $y = \lambda x$ and $z = \mu x$ we have
$$
a^2-c^2=x^2(1-\mu^2)\\
ab = x^2\lambda\\
ac = x^2\mu\\
bc = x^2\lambda\mu
$$
and making now $A = \frac ax,B = \frac bx, C = \frac cx$
$$
A^2-C^2=1-\mu^2\\
AB = \lambda\\
AC = \mu\\
BC = \lambda\mu
$$
so we conclude easily
$$
A^2B^2C^2=\lambda^2\mu^2\\
A^2-C^2= 1-A^2C^2\Rightarrow A^2=1\Rightarrow a^2=x^2\Rightarrow b^2= y^2\Rightarrow c^2= z^2
$$
