K. Adaricheva and M. Bolat have recently proved that if U 0 and U 1 are circles in a triangle with vertices A0, A1, A2, then there exist j ∈ {0, 1, 2} and k ∈ {0, 1} such that U 1−k is included in the convex hull of U k ∪ ({A0, A1, A2} \ {Aj}). 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. Grätzer and E. Knapp, lead to our result.