Helly-type theorems for homothets of planar convex curves

Helly's theorem implies that if $\boldsymbol{\mathcal{S}}$ is a finite collection of (positive) homothets of a planar convex body $B$, any three having non-empty intersection, then $\boldsymbol{\mathcal{S}}$ has non-empty intersection. We show that for collections $\boldsymbol{\mathcal{S}}$ of homothets (including translates) of the boundary $\partial B$, if any four curves in $\boldsymbol{\mathcal{S}}$ have non-empty intersection, then $\boldsymbol{\mathcal{S}}$ has non-empty intersection. We prove the following dual version: If any four points of a finite set $S$ in the plane can be covered by a translate [homothet] of $\partial B$, then $S$ can be covered by a translate [homothet] of $\partial B$. These results are best possible in general.