The area of the triangle with vertices A(z),B(iz) and C(z+iz) is

A(z)=x+iy˙=(x,y)B(iz)=-y+ix˙=(-y,x)z+iz=x-y+i(x+y)=(x-y,x-y) Area of the triangle =12|x(-y)-y(x)+(x-y)(y-x)|=12x2+y2=12z2