  Given strong local Dirichlet forms and $\mathbb{R}^N$-valued functions on a
metrizable space, we introduce the concepts of geodesic distance and intrinsic
distance on the basis of these objects. They are defined in a geometric and an
analytic way, respectively, and they are closely related with each other in
some classical situations. In this paper, we study the relations of these
distances when the underlying space has a fractal structure. In particular, we
prove their coincidence for a class of self-similar fractals.
