Equation of Circle in Complex Plane/Formulation 1

Theorem
Let $\C$ be the complex plane.

Let $C$ be a circle in $\C$ whose radius is $r \in \R_{>0}$ and whose center is $\alpha \in \C$.

Then $C$ may be written as:
 * $\cmod {z - \alpha} = r$

where $\cmod {\, \cdot \,}$ denotes complex modulus.

Proof
Let $z = x + i y$.

Let $\alpha = a + i b$.

Thus:

The result follows from Equation of Circle.