  We expose the relation between the properties of the three-body continuum
states and their two-body subsystems. These properties refer to their bound and
virtual states and resonances, all defined as poles of the $S$-matrix. For one
infinitely heavy core and two non-interacting light particles, the complex
energies of the three-body poles are the sum of the two two-body complex
pole-energies. These generic relations are modified by center-of-mass effects
which alone can produce a Borromean system. We show how the three-body states
evolve in $^6$He, $^6$Li, and $^6$Be when the nucleon-nucleon interaction is
continuously switched on. The schematic model is able to reproduce the main
properties in their spectra. Realistic calculations for these nuclei are shown
in detail for comparison. The implications of a core with non-zero spin are
investigated and illustrated for $^{17}$Ne ($^{15}$O+p+p). Dimensionless units
allow predictions for systems of different scales.
