Q.Show that the four points in the Argand Plane represented by the complex numbers $2+i$, $4+3i$, $2+5i$, $3i$ are the vertices of a square.

Plot the four points as $(x,y)$ pairs, show all four sides are equal and both diagonals are equal.

The points are $A=2+i\to(2,1)$, $B=4+3i\to(4,3)$, $C=2+5i\to(2,5)$, $D=3i\to(0,3)$.

Sides:

$AB=\sqrt{(4-2)^2+(3-1)^2}=\sqrt{4+4}=2\sqrt2,$

$BC=\sqrt{(2-4)^2+(5-3)^2}=\sqrt{4+4}=2\sqrt2,$

$CD=\sqrt{(0-2)^2+(3-5)^2}=\sqrt{4+4}=2\sqrt2,$

$DA=\sqrt{(2-0)^2+(1-3)^2}=\sqrt{4+4}=2\sqrt2.$

All four sides are equal, so $ABCD$ is at least a rhombus.

Diagonals:

$AC=\sqrt{(2-2)^2+(5-1)^2}=\sqrt{0+16}=4,$ …