the equation of circle in complex plane passing through three points I need hints on this question

Q.1 Show that the equation of circle passing through three points $z_1, z_2$ and $z_3$ is given by $$\displaystyle \frac{(z-z_1)/(z-z_2)}{(z_3-z_1)/(z_3-z_2)} = \frac{(\bar z-\bar z_1)/(\bar z-\bar z_2)}{(\bar z_3-\bar z_1)/(\bar z_3-\bar z_2)}$$  

 A: In $\mathbb{R}^2$, the circle passing through $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ passes satisfies a determinant:
\[ \left| \begin{array}{cccc} x^2 + y^2 & x & y & 1 \\\\
x^2_1 + y^2_1 & x_1 & y_1 & 1 \\\\  
x^2_2 + y^2_2 & x_2 & y_2 & 1 \\\\
x^2_3 + y^2_3 & x_3 & y_3 & 1 \\\\\end{array} \right| = 0  \]
This is considered a variant of Appolonius' problem of finding a circle tangent to 3 different circles.  Other problems are limiting cases, e.g. 


*

*Constructing a circle through a given point, tangent to a given line, and tangent to a given circle

*Finding the circles passing through two points and touching a circle
We can get a determinant formula for a circle with complex entries by setting $x = \frac{1}{2}(z + \overline{z}),\, y = \frac{1}{2}(z - \overline{z})\, x^2+y^2 = z \overline{z} = |z|^2$:
\[ \left| \begin{array}{cccc} |z\;|^2 & z & \overline{z} & 1 \\\\
 |z_1|^2 & z_1 & \overline{z}_1 & 1 \\\\
  |z_2|^2  & z_2 & \overline{z}_2 & 1 \\\\
  |z_3|^2  & z_3 & \overline{z}_3 & 1 \\\\
\end{array} \right| = 0  \]
This analytic formula hides the symmetries of the group of Möbius transformations [video] acting here.
A: *

*Start with three points a, b and c on the circle. 

*Define a linear fractional transformation by sending z to the cross ratio (z, a, b, c). Call it f(z).

*Note that this is invertible on the riemann sphere.

*For any point z, look at the complex conjugate of f(z), and apply the inverse of f to that point. Let the resultant point be w. Then we say that z and w are symmetric about your circle. 

*Try to prove that f sends your circle onto the real axis along with the point at infinity. 

*Conclude by noting that z lies on the circle iff f(z) is real or the point at infinity.

