This is also similar to a different technique for generating the [Sierpinski Triangle](http://en.wikipedia.org/wiki/Sierpinski_triangle) through an iterated method.

> Given any triangle $ABC$ with non-collinear endpoints $A,B,C$,
> select a random point $P_0$ within the center of $ABC$

> then iterate the following 

> pick any one of the vertices, $A,B,$ or $C$ at random

> generate $P_{i+1}$ as the midpoint of the line-segment $P_i$ and the randomly selected vertex

Iterated multiple times, this generates a Sierpinski triangle with a few extra points thrown in at the beginning.  If your initial point is definitely on the Sierpinski triangle (say you start with one of the vertices as your initial point $P_0$), then all of the subsequent points are definitely in the Sierpinski triangle.  The Sierpinski triangle has Hausdorff dimension $log(3)/log(2)$ ≈ $1.585$ (copied from wikipedia)

I remember writing this as a program in BASIC on the Apple ][, but I cannot recall the source of the question that led me to the program.  Most likely it was an article in Byte or Creative Computing.

This is similar to your selection a new point defined as the circumcenter of your three points, and then using that new point along with two of the vertices of your current triangle to generate then next triangle to find the circumcenter of. Your iterated method leads to points outside of the triangle, leading to a wandering triangle in most cases, leading to your second question, in which cases of initial triangles do you end up with convergence or divergence, which seems to have been addressed with some of the earlier answers.  I'll think about that a bit more before commenting on that.

This "iterated line-segment midpoint" technique definitely does not converge.  It leads to selecting points within the set of points contained within the Sierpinski Triangle.  The new points also do not diverge away; the points always stay within the confines of the triangle $ABC$, and if the initial point is in the Sierpinski triangle, then the set of points generated are all also within the set of points in Sierpinski triangle.