Let α, β, γ be three conics, one with foci B, C, one with foci C, A, and one with foci A, B. The specification of a single conic as the locus of the centre of a directed circle which touches two fixed directed circles is not unique but involves both a parameter and a choice of sign. Can we adjust the arbitrary elements in specifying α, β, γ to obtain a simple specification of the focus-sharing set as a whole? Explicitly, if α is derivable from circles η′, ζ′ round B, C, β from circles ζ′′, ξ′′ round C, A, and γ from circles ξ′′′, η′′′ round A, B, what conditions can bc usefully imposed on the concentric pairs ξ′′, ξ′′′; η′′′, η′; ζ′, ζ′′?