Let $A,B,$ and $C$ be three distinct fixed points. Let $D$ be a point that moves as follows:
I. Place $D$ anywhere.
II. Choose $A,B,$ or $C$ randomly.
III. Consider the mid-point of the line joining $D$ and the chosen point in step II.
IV. Move $D$ to the mid-point obtained in step III.
Repeat steps (II,III, and IV) large number of times.
You will end with a shape that is similar to the Sierpiński triangle, with the consideration where did you place $A,B,$ and $C$.
Here I have two questions;
Why does that happen?
and
If we repeat the number of iterations [steps II, III, IV] infinitely many times, then what is the ratio of the area where $D$ can not be there to the area of $\triangle ABC$? Assuming that $D$ was not initially placed inside $\triangle ABC$.
Any help will be appreciated.