Argand Digram when arguments are equal

Question

I am helping a student to draw the following equation on complex plane: $$\arg\left[z-\left(3+i\right)\right]=\arg\left[z-\left(1+3\,i\right)\right]$$

Can anyone explain why the dashed line to be neglected.

Answer 1

This equation gives two rays up and down on the line $x+y=4$, where points between (1,3) and (3,1) will not lie on this locus. For instance the points (2,2), (3/2,1/2), (1,1,3/2),... do not satisfy the equation that $$Arg[z-(3+i)]=Arg[z-(1+3i)].$$ For the point $2+2i$, we get LHS: $Arg(-1+i)=-3\pi/4$ but RHS: $Arg(1-i)=-\pi/4.$