# We cannot trisect with three circles and one additional line

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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