Abstract. Kira Adaricheva and Madina 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 $j\in\set {0,1,2}$ and $k\in\set {0,1}$ such that $U_{1-k}$ is included in the convex hull of $U_k\cup(\set{A_0,A_1, A_2}\setminus\set {A_j})$. We give a short new proof for this result, and we point out that a straightforward generalization for spheres fails.

DOI: 10.14232/actasm-016-307-7

AMS Subject Classification (1991): 52C99, 52A01

Keyword(s): convex hull, circle, sphere, abstract convex geometry, anti-exchange system, Carathéodory's theorem, carousel rule

Received October 8, 2016, and in revised form May 17, 2017. (Registered under 57/2016.)