<s>[INST] The image of a circle under $ z \mapsto \frac{z- \frac{1}{2}}{\frac{1}{2}z-1} $ Let $C_1 = \{ z: |z| = \frac{1}{7}  \}$ be a circle inside the unit circle $C_0 = \{ z: |z|=1\}$.  The fractional linear transformation:
$$ \phi: z \mapsto \frac{z- \frac{1}{2}}{\frac{1}{2}z-1} $$
is a map from the unit cirle to itself, $\phi: C_0 \to C_0$.  What is the image of $C_1$ ?  Obviously it is a circle, but what is the radius and center of $\phi(C_1)$?
I hope this figure not too confusing.  $\phi$ maps the light blue circle to the dark blue circle.  Similarly it maps light green to dark green.

In regard to 3 points... I know there is formula for 3 points $(x_1, y_1), (x_2, y_2),(x_3, y_3)$ going through a circle using a determinant but I have not used it:
$$ \left|\begin{array}{cccc} x^2 + y^2 & x & y & 1 \\ 
x_1^2 + y_1^2 & x_1 & y_1 & 1 \\
x_2^2 + y_2^2 & x_2 & y_2 & 1 \\
x_3^2 + y_3^2 & x_3 & y_3 & 1 \\ \end{array} \right|=0$$
Which 3 points should I pick anyway?  Even more alternatively, are there ways to solve this using power of a point from geometry?

I suspect @vvnitram's answer is wrong.  Here is a plot of the image circle (computed numerically) and the circle he proposes.  These are not quite the same.

 [/INST] Since at this point I am solving my own question, let me note that:
$$ \phi: \frac{1}{7} \mapsto  \frac{ \frac{1}{7}- \frac{1}{2}}{\frac{1}{2}\times\frac{1}{7}-1}, \hspace{0.25in}
-\frac{1}{7} \mapsto  \frac{ -\frac{1}{7}- \frac{1}{2}}{-\frac{1}{2}\times\frac{1}{7}-1}, \hspace{0.25in} 
\frac{i}{7} \mapsto  \frac{ \frac{i}{7}- \frac{1}{2}}{\frac{1}{2}\times\frac{i}{7}-1}
  $$
Those are fine and then I have to solve the determinant equation:
$$ \left|\begin{array}{cccc} x^2 + y^2 & x & y & 1 \\ 
x_1^2 + y_1^2 & x_1 & y_1 & 1 \\
x_2^2 + y_2^2 & x_2 & y_2 & 1 \\
x_3^2 + y_3^2 & x_3 & y_3 & 1 \\ \end{array} \right|=0 $$
Here's re-write in terms complex numbers $z = x+iy$ and $z = x - iy$.  I did not to the change of basis, I just am guessing:
$$ \left|\begin{array}{rlll} |z|^2 & z & \overline{z} & 1 \\ 
|z_1|^2 & z_1 & \overline{z}_1  & 1 \\
|z_2|^2 & z_2 & \overline{z}_2 & 1 \\
|z_3|^2 & z_3 & \overline{z}_3 & 1 \\ \end{array} \right|=0 $$
In my case, $z_1 = \phi(\frac{1}{7})$ and $z_1 = \phi(-\frac{1}{7})$ and $z_1 = \phi(\frac{i}{7})$ .
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