Condition for Points in Complex Plane to form Isosceles Triangle/Examples/1+2i, 4-2i, 1-6i

Examples of Use of Condition for Points in Complex Plane to form Isosceles Triangle
Let $A = z_1 = 1 + 2 i$, $B = z_2 = 4 - 2 i$ and $C = z_3 = 1 - 6 i$ represent on the complex plane the vertices of a triangle.

Then $\triangle ABC$ is isosceles, where $B$ is the apex.

Proof
By Condition for Points in Complex Plane to form Isosceles Triangle:


 * $\triangle ABC$ is isosceles, where $A$ is the apex, $AB = AC$.

Hence:

Similarly:

So $AB = BC$ and so $\triangle ABC$ is isosceles.

Finally note that:

demonstrating that $\triangle ABC$ is not equilateral.