Why are there seven circles here, when Apollonius says eight?
The drawing is illustrative. Every centre and radius is read from the certificate — no coordinate is entered by hand. Colour is the antipodal class; solid is the first circle of that class and dashed the second, ordered by radius. A class can hold two circles carrying the same mode label — each circle is labelled by whichever of the two complementary modes gives it a positive radius, and both can fall on one side.
Exact computed solution
COMPUTEDWatch the eighth circle leave
Raise d towards the locus and the second circle of the (o,i,i) class grows without bound. At d = 2√2 it is the line. Past the locus it comes back from infinity — carrying the other mode label of its class.
The sweep
COMPUTEDWhere the count comes from
Eight modes, four antipodal classes. Each class contributes two signed roots — unless its quadratic is not quadratic. Select a row to see its coefficients.