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. 
 

$$\begin{gathered} A\left(2, 1\right)B\left(4, 3\right),C\left(2, 5\right), D\left(0, 3\right) \\ AB=\sqrt{{\left(4-2\right)}^2+{\left(3-1\right)}^2}=2\sqrt{2} slope \\ AB=\frac{3-1}{4-2}=1 \\ BC=\sqrt{{\left(4-2\right)}^2+{\left(3-1\right)}^2}=2\sqrt{2} slope \\ BC=\frac{5-3}{2-4}=-1 \\ CD=\sqrt{{\left(2-0\right)}^2+{\left(5-3\right)}^2}=2\sqrt{2} AB\perp BC \\ DA=\sqrt{{\left(0-2\right)}^2+{\left(3-1\right)}^2}=2\sqrt{2} \\ \therefore A, B, C, D from a square \end{gathered}$$