Let $$S = \sqrt{x^2+c^2+2x\left(\frac{y}{z}(c+d)-d\right)}$$ and $$B= \frac{S-x+d}{\frac{y}{z}{(c+d)}}, $$ we are supposed to find $d$ (which is independent of x) s.t.: $$ B^2y+B\left(x\left(1-\frac{1}{S}\right)-z\right)-x\left(1-\frac{1}{S}\right)>0.$$ It is given that $z>y>0$, $x>0$, $c\geq z+1$, $d>z-1$, all real valued.
I would say that there must be some clever substitution under which one can solve the discriminant equal to zero question, however could not find any (which can be solved). Direct attempt yields messy equation with a lot of square roots.