Condition for Points in Complex Plane to form Parallelogram/Examples/2+i, 3+2i, 2+3i, 1+2i

Theorem
The points in the complex plane represented by the complex numbers:
 * $2 + i, 3 + 2 i, 2 + 3 i, 1 + 2 i$

are the vertices of a square.

Proof
Let us label the points:

From Geometrical Interpretation of Complex Subtraction, we have that the difference $p - q$ between two complex numbers $p, q$ represents the vector $\vec {q p}$.

Let us take the differences of the complex numbers given:

So:
 * $\vec {AB} = \vec {DC} = 1 + i$
 * $\vec {DA} = \vec {CB} = 1 - i$

Thus by definition $ABCD$ forms a parallelogram.

Next it is noted that:
 * $\paren {1 + i} i = i + i^2 = 1 - i$

and so $AB$ and $DC$ are perpendicular to $DA$ and $CB$.

Thus by definition $ABCD$ is a rectangle.

Finally note that:
 * $\cmod {1 + i} = \cmod {1 - i} = \sqrt {1^2 + 1^2} = \sqrt 2$

and so:
 * $\cmod {AB} = \cmod {DC} = \cmod {DC} = \cmod {DC}$

That is, all four sides of $ABCD$ are the same length.

Thus by definition $ABCD$ is a square.