Here's what you got, x²+y²+z²-xy-yz-zx=0. Multiply this equation with 2. You get this, 2x²+2y²+2z²-2xy-2yz-2zx=0. Rearrange the terms in this manner, (x²-2xy+y²) + (y²-2yz+z²) + (z²-2zx+x²)=0. You have, (x-y)²+(y-z)²+(z-x)²=0. Here's a little brain work you need to do.
You might be knowing that the square of a real number is either zero or positive. Now, see that the terms on the L.H.S. are all squares of real numbers ( if you think of x, y, z to be real numbers instead of complex, which you forgot to mention ) and the R.H.S. is 0. Hence, each term has to be 0, if this were not so the L.H.S. would be positive. So you have, x-y=0 and y-z=0 and z-x=0 giving you x=y=z.