We have seen a trisecting construction using seven circles.
We prove six circles is not enough by using a Maple program that creates all
points that can be constructed with a given number of circles (Maple file,
pdf). Essentially, we construct
all four-circle constructions (there are 14 up to symmetry) and note that none
of these go through the point (1/3, 0) or (-1/3, 0). 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.
We summarize by giving the number of points generated by N circles by our
Mohr-Mascheroni construction.
For convenience, 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 points
N = 3 circles: 2 new points
N = 4 circles: 11 new points
N = 5 circles: 300 new points
A list of all points that can be constructed by 2, 3, 4, or 5 circles is in this Maple file (pdf).