Given an open ball $B=B(\xi,r) \subset \mathbb{R}^3$, and two points $x,y \notin B$, how long (by Euclidean metric) is the shortest path from $x$ to $y$ that does not intersect $B$?
I am in a context where a reference is much more useful than a proof.
By considering a plane which includes $x$, $y$ and $\xi$, the problem reduces to a problem of avoiding a circle in $\mathbb{R}^2$. I guess this can be found as an exercise in an elementary geometry book, but that literature is very unfamiliar to me.