As a suggestion, you could re-write x,y,z as x* y* z* where x* = x + a, y* = y + b and z* = z + c and then simplify in terms of a, b, and c on one side. If you can prove that they have to be zero then you are done.
Also note that when you have squared terms, you know that things have to be positive.
A starting suggestion though would be firstly to get rid of the denominator on both sides by cross multiplying terms between the left and right hand sides.
In addition to this if you need more tools, take a look at the properties of norms if you have to. I don't think you will need to but norms are used with questions like this.