You decide to go for a walk in a garden in the shape of a right isosceles triangle, ABC; see the figure on the left.  Starting at the vertex A, you walk to D, the midpoint of the BC; from D to E, the midpoint of AC; from E to F, the midpoint of AD; from F to G, the midpoint of DE; and so on...forever!  At what point in the garden do you end up?  (That is, if there exists one, what is the limiting point?)

For further thought:  How far did you walk?