Eight Circles, Except When There Are Seven

Apollonius' problem has eight solutions in generic position. While formalizing it in MachLib we assumed the obvious general-position condition — three equal circles, comfortably separated. That assumption is false. At the exact locus d² = 8ρ² one of the four solution classes loses its leading coefficient, its quadratic becomes linear, and the count drops to seven. Nothing in the picture…