We settle a number of questions
concerning definability in first order logic
with an extra predicate symbol ranging over semi-linear sets.
We give new results both on the positive and
negative side: we show that in first-order logic one cannot query
a semi-linear set as to whether or not it contains a line, or
whether or not it contains the line segment between two given
points. However, we show that some of these queries become
definable if one makes small restrictions on the semi-linear sets
considered.