  A special case of a theorem of De Bruijn and Erd\H{o}s asserts that any
noncollinear set of $n$ points in the plane determines at least $n$ distinct
lines. Chen and Chv\'atal conjectured a generalization of this result to
arbitrary finite metric spaces, with a particular definition of lines in a
metric space. We prove it for metric spaces induced by connected
distance-hereditary graphs -- a graph $G$ is called distance-hereditary if the
distance between two vertices $u$ and $v$ in any connected induced subgraph $H$
of $G$ is equal to the distance between $u$ and $v$ in $G$.
