This article presents a simple yet effective method for computing geodesic distances on triangle meshes. Unlike the popular window propagation methods that partition mesh edges into intervals of varying lengths, our method places evenly-spaced, source-independent Steiner points on edges. Given a source vertex, our method constructs a Steiner-point graph that partitions the surface into mutually exclusive tracks, called geodesic tracks. Inside each triangle, the tracks form sub-regions in which the change of distance field is approximately linear. Our method does not require any pre-computation, and can effectively balance speed and accuracy. Experimental results show that with 5 Steiner points on each edge, the mean relative error is less than 0.3 % for common 3D models used in the graphics community. We propose a set of effective filtering rules to eliminate a large amount of useless broadcast events. For a 1000K-face model, our method runs 10 times faster than the conventional Steiner point method that examines a complete graph of Steiner points in each triangle. We also observe that using more Steiner points increases the accuracy at only a small extra computational cost. Our method works well for meshes with poor triangulation and non-manifold configuration, which often poses challenges to the existing PDE methods. We show that geodesic tracks, as a new data structure that encodes rich information of discrete geodesics, support accurate geodesic path and isoline tracing, and efficient distance query. Our method can be easily extended to meshes with non-constant density functions and/or anisotropic metrics.