Cross ratio on $T(z)=(z,2i;2,-2)$ and $S(w)=(w,-1;2i,1+4i)$

Question

Cross ratio on $T(z)=(z,2i;2,-2)$ and $S(w)=(w,-1;2i,1+4i)$. I am using these to find a Mobius transformation that takes the circle $|z|=2$ to the line $2x-y=2$. I want to map the points $u=2i, v=2$, and $w=-2$ on the circle to the points $x=-1, y=2i$, and $q=1+4i$ on the line, so $u→x$, $v→y$ and $w→q$. So, I'm setting $T(z)=S(w)$ which involves doing the cross ratio on both sides and then solving for $w$. I tried this and my computation got extremely messy and now I don't know where to go from the very long equation that doesn't seem right.