= Four contact points of a circle chain
Four positive-<radius> <circles> with pairwise disjoint interiors, successive <external tangencies>, and four distinct contact points have those contact points on a <circle>. Apply the reciprocal <Möbius transformation> $w=1/(z-T)$ at one contact $T$. The two <circles> through $T$ become disjoint <parallel lines>; the other two <disks> lie between them and avoid zero. <Collinearity for tangent circles between parallel lines> places the three finite transformed contacts on a <straight line>. That <straight line> misses zero: otherwise its segment across the strip would pass through zero, while that segment is contained in the two transformed closed <disks>. Its inverse is therefore a genuine <circle> through $T$, containing all four original contacts. Allowing <internal tangency> without the pictured disjoint-interior hypotheses admits collinear degeneracies instead.
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