It is known that A δ-hyperbolic space is a geodesic metric space in which every geodesic triangle is δ-thin. (http://en.wikipedia.org/wiki/%CE%94-hyperbolic_space)

The question is that if we remove the word "geodesic" from the above definition, is it still acceptable? Or does the sentence a δ-hyperbolic space is a metric space in which every geodesic triangle is δ-thin still make sense?

Many thanks!