$\displaystyle x, y, z$ are all different real numbers.

When they satisfy the equation: $\displaystyle x^2-kyz=y^2-kzx=z^2-kxy$,

find the value of $\displaystyle x+y+z+k$.

I already found the value of $\displaystyle k$ though; it's $\displaystyle -2$.

But I can't seem to go any further from that.