Chaos Game with a twist

The traditional triangular so called "chaos game" involves choosing three points on the plane, defining a triangle. One then starts from any location inside the triangle, and move to a location half way towards one of the triangle vertices. The process is repeated from this new location. Each location in this process is drawn as a small marker. The result is shown as the first image here, this is the attractor for the process since any starting location will result in the same image.

The variation here is to choose many starting locations outside the triangle and draw the path the series takes into the interior attractor.