On proving that $\mathbb{Z}[\omega]$ is a Euclidean domain.

Question

To connclude the proof of an Eisenstein integer to be an Euclidean domain I need to show that $N(r)<N(\beta)$ where I have this assumption $|t-p|\leq\frac{1}{2}$ and $|s-q|\leq\frac{1}{2}$, $r=\beta \phi$ where $\phi=(t-p)+(s-q)\omega$ and $t=\frac{ac+bd-ad}{c^2+d^2-cd}$ and $s=\frac{bc-ad}{c^2+d^2-cd}$ where $a,b,c,d \in \mathbb{Z}$. Then $N(r)= N(\beta \phi)$$=N(\beta)N(\phi)$ $=N(\beta)((t-p)^2+(s-q)^2-(t-p)(s-q))\leq N(\beta)(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}) =\frac{3}{4}N(\beta)<N(\beta).$ Am I allowed to compute $N(\phi)$ where $t-p$ and $s-q$ are not surely integers ? If it is not integers then it violates that definition of the norm of Eisenstein integers. Is $t-p$ and $s-q$ are inetegers ?