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.

The four points form a quadrilateral with equal sides $\sqrt8$ and equal diagonals $4$, hence a square.

The complex numbers correspond to $A(2,1),\ B(4,3),\ C(2,5),\ D(0,3)$ in the Argand plane.

Side lengths:

$AB=\sqrt{(4-2)^2+(3-1)^2}=\sqrt{8}$,

$BC=\sqrt{(2-4)^2+(5-3)^2}=\sqrt{8}$,

$CD=\sqrt{(0-2)^2+(3-5)^2}=\sqrt{8}$,

$DA=\sqrt{(2-0)^2+(1-3)^2}=\sqrt{8}$.

So all four sides are equal, i.e. $ABCD$ is a rhombus.

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