In the applet above the white triangle corresponds to the set of all possible triangles. By clicking on the arrows you can change the depth of the recursion (under the Gergonne point recursion). Clicking on the box "Add more points" will plot several thousand points under the Gergonne point recursion. This is done by randomly generating a triangle and then taking a random walk of the depth and plotting the point, then repeating the process a few thousand times.
After clicking you can move thr cursor in the triangle and the triangle corresponding to your current position will be shown on the left in orange.