Equation of Circle in Complex Plane/Formulation 2

Theorem
Let $\C$ be the complex plane.

Let $C$ be a circle in $\C$.

Then $C$ may be written as:
 * $\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$

where $\alpha$ and $\gamma$ are real and $\beta$ may be complex.

Proof
From Equation of Circle in Cartesian Plane, the equation for a circle is:


 * $A \left({x^2 + y^2}\right) + B x + C y + D = 0$

Thus:

The result follows by setting $\alpha := A$, $\beta := \dfrac B 2 + \dfrac C {2 i}$ and $\gamma := D$.