Here's a cute question I came up with.
Start with a circle, and then choose three points $a$, $b$, and $c$ on the circle, and proceed as follows:
- Draw the triangle inside the circle with vertices $a$, $b$, and $c$
- Draw the inscribed circle of that triangle, which is tangent to each of the three sides of triangle, and now label these three points of tangency as $a$, $b$, and $c$ (so we're updating which points we're calling points $a$, $b$, and $c$).
- Repeat from Step 1.
This construction gives us a sequence of inscribed circles and triangles that telescope down to a point, a limit point, which is determined only by the initial choice of the points $a$, $b$, and $c$. Which points in the interior of the circle are limit points of this construction?
Also, if anyone has ideas for more interesting variations of this question, I'd like to hear them.