Let the points be z(x,y),z1(x1,y1) and z2(x2,y2) then, z−z1=x+iy−x1−iy1 =x−x1+i(y−y1) And, z−z2=x+iy−x2−iy2 =x−x2+i(y−y2) Now, z−z1z−z2=
[(x−x1)+i(y−y1)]
[(x−x2)+i(y−y2)]
×
[(x−x2)−i(y−y2)]
[(x−x2)−i(y−y2)]
=
i[(x−x2)(y−y1)−(x−x1)(y−y2)]
(x−x2)2+(y−y2)2
since, arg(
z−z1
z−z2
)=0 or π then [
[(x−x2)(y−y1)]−[(x−x1)(y−y2)]
t[(x−x1)(x−x2)]+[(y−y1)(y−y2)]
]=0 (x−x2)(y−y1)=(x−x1)(y−y2) xy−xy1−x2y+x2y1=xy−xy2−x1y+x1y2 x(y2−y1)+y(x1−x2)+(x2y1−x1y2)=0 It represents the equation of a straight line passing through A and B.