\[| z - w \| = \|z-u \|\]
is the complex form of a line. To get the Cartesian form of the line let \[z=x+iy, \: w=a+bi, \; u=c+di\]
then\[| (x+iy)-(a+bi) \| = \|(x+iy)-(c+di) \|\]
\[| (x-a)+(y-b)i \| = \|(x-c)+(y-d)i \|\]
\[ (x-a)^2+(y-b)^2 = (x-c)^2+(y-d)^2\]
\[ x^2-2ax+a^2+y^2-2by+b^2 = x^2-2cx+c^2+y^2-2yd+d^2\]
\[-2ax+a^2-2by+b^2 =-2cx+c^2-2yd+d^2\]
\[a^2+2yd-2by+b^2 =2ax-2cx+c^2-2yd+d^2\]
\[2yd-2by =2ax-2cx-2yd-a^2-b^2+c^2+d^2\]
\[2y(d-b) =2(a-c)x-a^2-b^2+c^2+d^2\]
\[y =\frac{a-c}{d-b}x+ \frac{-a^2-b^2+c^2+d^2}{2(d-b)}\]