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

TL;DR

This paper provides a concise new proof of a geometric theorem involving circles within a triangle, highlighting its simplicity and discussing the limitations of extending the result to spheres.

Contribution

The authors present a simplified proof of an existing theorem on circle inclusion in triangles and analyze why this approach does not extend straightforwardly to spheres.

Findings

New concise proof of the circle theorem

Identification of limitations when generalizing to spheres

Clarification of geometric inclusion properties

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.