If we want to talk about geometry in a metric space, we go through the following procedure to define geodesics. We define the length of a path in the (complete) space as the length given by integrating over infinitesimal distances on the path. The intrinsic metric of the space is then given by the infimum of the lengths of all paths between two points. If the intrinsic metric agrees with the metric we call the space a 'length space'; If furthermore there is a unique path between any two points whose length is the same as the distance between the points, we call that path the geodesic between the two points. We can from here go on and talk about geometry.
The above definition uses that property of straight lines in the euclidean plane, that they are the shortest paths between two points, to generalize to abitrary metric spaces. This is fine and also nicely connects with the usual definition of geodesics in Riemannian geometry, however it's only applicable in a restricted class of metric spaces (length spaces with unique geodesics), and seems to rely on further analytic properties instead of propeties related only to the metric itself.
However, here is an alternative defintion of straight lines in general metric spaces that seems to me much more general, applicable to many more metric spaces and also relies only on simple properties of of the metric. A further property of straight lines in the euclidean plane, besides the fact that they form the shortest path between two points, is that they are the locus of all points equidistant from two given points. Let us, then, define a straight line in a metric space as the locus of all points equidistant from two given points; I played with some equations and it seems to me that at least in taxicab space it gives the 'correct' lines. What is the problem with this definition? Why is it not used? Which properties of straight lines and the geometry are generalized to general metric spaces if we use this defintion, and which properties are lost?