  K. Adaricheva and M. Bolat have recently proved that if $U_0$ and $U_1$ are
circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in
\{0,1,2\}$ and $k\in\{0,1\}$ such that $U_{1-k}$ is included in the convex hull
of $U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. One could say disks instead of
circles. Here we prove the existence of such a $j$ and $k$ for the more general
case where $U_0$ and $U_1$ are compact sets in the plane such that $U_1$ is
obtained from $U_0$ by a positive homothety or by a translation. Also, we give
a short survey to show how lattice theoretical antecedents, including a series
of papers on planar semimodular lattices by G. Gratzer and E. Knapp, lead to
our result.
