Iterations in Geometry, Generalization

Iterations that start with a point in the plane of ABC and move first half way to vertex A, and from there half way to vertex B, and then half way to vertex C, and so on, converge to a triangle defined by three points

This process can be generalized in two ways. First, on every step, the distance to a subsequent vertex could be cut in a ratio R_n:R_d different from 1:1. It is easy to see that if

then the iterations converge to

where s = 1 + a + a². If D is the first of the three points in (2), then the cevian fromA through D, cuts BC in the ratio a:1.

Secondly, the process can apply to any N-gon, not necessarily a triangle. (2) has a natural generalization for N vertices. In the applet below, three numbers N, R_n, and R_d are all modifiable. Clicking a little off but to the right of the number's vertical center line will increase the number, clicking to the left will decrease it.