To find this set of points, we construct two circles of equal radius centered at A and B. We then construct two more circles with equal radii greater than the radii of the first two circles by some constant.

The points of interest are labeled in the drawing as D, E, F, and G. Let us call the radii of the small circles R and the radii of the large circles R+k, where k is a constant. Then the points D and G are both R away from point A and R+k away from B. Likewise, the points E and F are both R away from B and R+k away from A. Therefore, these four points will always have distances from A and B with a difference of k, as R increases and decreases.

To use the drawing, move C to change the radius of the inner circles. Move A or B to change the distance between the foci. Click on the red X in the bottom of the applet to clear the trace marks.