How to find whole number answers in systems of square root equations Given the following 4 equations, can you find 4 whole number answers using whole number variable inputs?
$x,y,z$ where $x>y>z$ 
$Eq 1 = (x^2-2xy+y^2-2xz+z^2)^{\frac{1}{2}} $
$Eq 2 = (x^2-2xy+y^2+2xz-z^2)^{\frac{1}{2}} $
$Eq 3 = (x^2+2xy-y^2+2xz-z^2)^{\frac{1}{2}}$
$Eq 4 = (x^2+2xy-y^2-2xz+z^2)^{\frac{1}{2}} $
for example, $x=1921, y=792, z=272$ yields $551$ in equation 1, but non-whole number answers for remaining 3 equations

Do the substitution $x=c,\,$ $y=x-a,\,$ $z=x-b,\,$ and we get the system,
$$a^2+b^2-c^2 = s_1^2\tag1$$
$$a^2-b^2+c^2 = s_2^2\tag2$$
$$-a^2-b^2+3c^2 = s_3^2\tag3$$
$$-a^2+b^2+c^2 = s_4^2\tag4$$
Solving for $(1)$ as $u_1^2+u_2^2=v_1^2+v_2^2,\,$ and the other three are just quadratic forms to be made squares. Then solving simultaneously any two of the remaining three involves an elliptic curve, so there are infinitely many solutions to the $3$-out-of-$4$ system. In fact, there is a polynomial solution to $(1), (2),(4)$, a "small" particular example being,
$$x=5405,\; y=4440,\; z=22$$
However, I haven't found a solution for all four.
