The area of the triangle on the argand plane formed by the complex numbers $-z,iz,z-iz$ , is (where $i=\sqrt{-1}$ ):

Let $z=x+iy$ , then coordinates in ordered pair of the points corresponding the complex numbers $-z$ , is and $z-iz$  are $\left(-x,-y\right),\left(-y,x\right)$  and $\left(x+y,y-x\right)$ , respectively.

Hence area of triangle $=\frac{1}{2}\left|\left|\begin{array}{ccc}-x& -y& 1\\ -y& x& 1\\ x+y& y-x& 1\end{array}\right|\right|$

$=\frac{1}{2}\left|\left|\begin{array}{ccc}-x& -y& 1\\ -y& x& 1\\ 0& 0& 3\end{array}\right|\right|$ $\left(R_2\to R_3+\left(R_1+R_2\right)\right)$

Expand w.r.t. $R_3$

$=\frac{3}{2}\left|\left|\begin{array}{cc}-x& -y\\ -y& x\end{array}\right|\right|$

$=\frac{3}{2}\left(x^2+y^2\right)$

$=\frac{3}{2}{\left|z\right|}^2$