Iterated triangle partitions

For a given triangle there are many points associated with the triangle that lie in its interior; examples include the incenter (which can be found by the intersection of the angle bisectors) and the centroid (which can be found by the intersection of the medians). Using this point one can naturally subdivide the triangle into either three or six “daughter” triangles. We can then repeat the same process on each of the daughters and so on and so on. A natural question is after some large number of steps what does a typical nth generation daughter look like (up to similarity)? In this paper we look at this problem for both the incenter and the centroid and show that they have very distinct behavior as n gets large. We will also look at the Gergonne point and the Lemoine point.