Let $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ be the unitary complex disk. Given $z_1,z_2\in \mathbb{D}$, for every $z\neq 1$ show that in the closed triangle $1,z_1,z_2$, we have $$\dfrac{|1-z|}{1-|z|}<K,$$ where $K$ is a constant only depending on $z_1$ and $z_2$.
My attempt:
I tried to consider the line passing through $1$ and $z$ and the intersection point of such line with the line joining $z_1$ with $z_2$ and try to get relations between $z$ and this line, but I couldn't figure out how to get the constant $K$. Any suggestion?
Thanks in advance!
