Given distinct planar points $A$ and $B$ and a real number $k$, called the index, Apollonius long ago showed that the locus of all points $P$ for which $k$ is the ratio of the distances from $P$ to $A$ and from $P$ to $B$ is a circle. The puzzle we present is this one: Given a circle $\mathcal Q$ in the hyperbolic disk whose diameter $CD$ lies along the real axis, how may we recover $\mathcal Q$ as a circle of Apollonius? That is, what are $A$, $B$, and $k$? The beauty of this puzzle is that $B$ is the hyperbolic center of $\mathcal Q$.

This talk topic lies under the category of MSC 51M09: elementary problems in hyperbolic geometry. The talk is for a general audience, including undergraduate mathematics majors.