The area of the triangle on the complex plane formed by the complex numbers z, -iz and z+ i z is

Let $z=x+iy$

Then $-iz=-i(x+iy)=y-ix$

$z+iz=(x+iy)+i(x+iy)=(x-y)+i(x+y)$

Area of triangle with vertices $A(x_1,y_1) , B(x_2,y_2) ,C(x_3,y_3)$$=\frac{1}{2}\left[x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right]$

$=\frac{1}{2}\left[x\left(-2x-y\right)+yx+\left(x-y\right)\left(y+x\right)\right]$

$=\frac{1}{2}\left(x^2+y^2\right)$

$=\frac{1}{2}{\left|z\right|}^2$