An easy way to a theorem of Kira Adaricheva and Madina Bolat on convexity and circles

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