Beginning with points A and B and the line through them, we construct the circle with center at B and radius to A. There are only two ways, up to symmetry, to construct a second circle.
The following outlines all the possibilities for three circles and a line. We use symmetry whenever possible; for instance, we only need to consider third circles with centers in the "first quadrant" (E, D or C) for the case of the two congruent circles. We only draw lines from points in the top half to points in the bottom half since a line through an existing point on the x-axis does not create a new point on the x-axis.
We assume that B is at the origin and that A is at -1 and C is at 1 on the x-axis. Since we could equally well consider A or C to be the origin, we only consider the constructible values on the x-axis modulo 1, hence only need to list values between -0.5 and 0.5. Since we can flip the diagrams, we also can consider the constructible values on the x-axis modulo the +/- sign, hence we only need to list values between 0 and 0.5. For instance, in the third 3-circle-1-line construction below which adds the circle with radius DB, the point M is actually at x = 2.6180339887 which modulo 1 is -0.3819660113 which modulo the +/- sign is 0.3819660113. This happens to be the value of K as well.
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