  A set of n points in the plane which are not all collinear defines at least n
distinct lines. Chen and Chv\'atal conjectured in 2008 that a similar result
can be achieved in the broader context of finite metric spaces. This conjecture
remains open even for graph metrics. In this article we prove that graphs with
no induced house nor induced cycle of length at least~5 verify the desired
property. We focus on lines generated by vertices at distance at most 2, define
a new notion of ``good pairs'' that might have application in larger families,
and finally use a discharging technique to count lines in irreducible graphs.
