Stolz region

[B บ ....]

Given two points $z_1, z_2 \in \mathbb{C}$ such that $|z_1| < 1$ and $|z_2|<1$, there exists $K > 0$ such that for any points $z \neq 1$ in the closed triangle with vertices $z_1, z_2,$ and 1,
$$\frac{|1-z|}{1-|z|} \leq K.$$
Find the smallest possible value of $K$ if $z_1 = \frac{1+i}{2}, z_2 = \frac{1-i}{2}.$

I let $z = x + iy$ in the closed triangle and use two dimensional calculus to find the absolute maximum on the closed triangle region. I find that the smallest possible $K$ is 1 in this case(Actually, it is at the point $z= \frac{1}{2} + i \frac{1}{2})$. I am not sure if this is true since the closed triangle region is not closed, so not compact. So I am not sure the method of finding absolute maximum in multivriate calculus can be applied here.

Can anyone recommend ?