Condition for Points in Complex Plane to form Isosceles Triangle

Theorem
Let $A = z_1 = x_1 + i y_1$, $B = z_2 = x_2 + i y_2$ and $C = z_3 = x_3 + i y_3$ represent on the complex plane the vertices of a triangle.

Then $\triangle ABC$ is isosceles, where $A$ is the apex, :
 * ${x_2}^2 + {y_2}^2 - 2 \paren {x_1 x_2 + y_1 y_2} = {x_3}^2 + {y_3}^2 - 2 \paren {x_1 x_3 + y_1 y_3}$

Proof
By definition of isosceles triangle:
 * $\triangle ABC$ is isosceles, where $A$ is the apex, $AB = AC$.

Hence: