More Than Six Circles Are Needed for Mohr-Mascheroni

We have seen a trisecting construction using seven circles. 

The proof that six circles is not enough is contained in this Maple file.  Essentially, we construct all four-circle constructions (there are 14 up to symmetry). We easily verify that if a six circle construction exists, the next two circles added must go through the desired trisecting point.  It is then easy to verify that no fifth circle goes through the desired (1/3,0) or (-1/3,0) point, hence one requires at least seven circles. 

A list of all points that can be constructed by 2, 3, 4, or 5 circles is in this Maple file. 

We summarize by giving the number of points generated by N circles by our Mohr-Mascheroni construction. 

We assume we begin with the two points, (1,0) and (-1,0).  By symmetry, we need only list those points in the first quadrant.  Here are the number of new points in the first quadrant generated by N circles:

N = 2 circles: 2 new points

N = 3 circles: 2 new points

N = 4 circles: 11 new points

N = 5 circles: 300 new points

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