Trouble with deducing the contradiction

[tt2348]

Let x,y,z be integers satisfying a specific condition, which boils down to
5|(x+y-z) and 2*5^{4}k=(x+y)(z-y)(z-x)((x+y)^2+(z-y)^2+(z-x)^2)
or equivalently 5^{4}k=(x+y)(z-y)(z-x)((x+y-z)^2-xy+xz+yz)
I want to show that GCD(x,y,z)≠1, starting with the assumption 5 dividing (x+y), (z-y), or (z-x) results in x,y or z being divisible by 5. then it's easy to show that 5 divides another term, implying 5 divides all three.
I run into trouble assuming 5 divides the latter part, 2((x+y)^2+(z-y)^2+(z-x)^2)=((x+y-z)^2-xy+xz+yz) and showing the contradiction from that point.
Any hints?

 

[haruspex]

Suppose 5 | a²+b²+c². Each term individually must be 0, 1 or 4 mod 5. The only ways for three such to add to 0 mod 5 involve at least one of them being 0 mod 5, no?