Why does the Chaos Game generate a fractal?

Someone hinted the answer to me, but deleted it. This person was absolutely correct, only the answer was incomplete (as it was a hint). This person said:

When you perform the chaos game, you actually perform transformations $T_1,T_2$, and $T_3$.

Here is why this is correct: the transformations for the Sierpinski triangle are: \begin{align*} T_1(x,y)&=(x/2,y/2)\\ T_2(x,y)&=(x/2+1/2,y/2)\\ T_3(x,y)&=(x/2+1/4,y/2+\sqrt{3}/4)\\ \end{align*}

And when you iteratively move half-way between the current point $(x,y)$ and a corner $(X_i,Y_i)$ of the triangle , you get the point with coordinates: $$ (X_i+\frac{1}{2}(x-X_i),Y_i+\frac{1}{2}(x-Y_i)) $$

If we replace $X_i,Y_i$ with the coordinates of the corners, we end up with exactly $T_1,T_2,T_3$.