In the applet above the white triangle corresponds to the set of all possible triangles, the blue and red X the current base triangle (the triangle(s) that this corresponds to is indicated on the right). The corresponding colored dots are the daughters. In the upper left you can choose a subdivision method (now allow for subdividing into 3 parts as well). You can plot one or two sets of points, if you plot two you can draw lines between corresponding daughters (if this last option is chosen the ratio of the distance between daughters and the original triangles is also shown).
By clicking in the main triangle and dragging you can see what happens to the behavior as the base triangles are changed. (Note when drawing two points clicking and dragging will effect the one nearest to where you first clicked.)