Equation of Circular Arc in Complex Plane

Theorem
Let $a, b \in \C$ be complex constants representing the points $A$ and $B$ respectively in the complex plane.

Let $z \in \C$ be a complex variable representing the point $Z$ in the complex plane.

Let $\lambda \in \R$ be a real constant such that $-\pi < \lambda < \pi$.

Then the equation:
 * $\arg \dfrac {z - b} {z - a} = \lambda$

represents the arc of a circle with $AB$ as a chord subtending an angle $\lambda$ at $Z$ on the circumference.

Proof

 * Circular-Arc-in-Complex-Plane.png

By Geometrical Interpretation of Complex Subtraction:
 * $z - a$ represents the line from $A$ to $Z$
 * $z - b$ represents the line from $B$ to $Z$

Thus:
 * $\arg \dfrac {z - b} {z - a} = \lambda$

represents the statement that the angle between $AZ$ and $BZ$ is constant:
 * $\angle AZB = \lambda$

That is, the angle subtended by $AB$ at $Z$ is $\lambda$.

The result follows from the Inscribed Angle Theorem.