In the applet above the white triangle corresponds to the set of all possible triangles, the blue "X" corresponds to the base triangle (i.e., the triangle the recursion starts with), this triangle is also shown on the left in yellow with angles indicated below. The red dots are the daughters of the triangle formed by using the indicated recursion method. The green plusses are similarly the daughters of the triangle formed by using the indicated recursion method.
By pushing the arrows you can change the recursion method to one of (bisectors, medians, Lemoine point, Gergonne point, Spieker center). You can also change the initial triangle by clicking or clicking and dragging inside the white triangle. The triangle shown on the right in orange (may take a few clicks to appear) corresponds to the triangle at your current cursor position.