This construction illustrates the five-circle theorem.
Starting from 5 points on the red circle, we make a pentagon.
The thin blue circles are constructed to contain the three vertices
of each of the 5 star points. They intersect in 5 other points; according
to the theorem these should be on a circle (drawn blue).
You may drag any of the red points on the red circle to check the
veracity of the theorem.
The strange alternative contructions you have experienced lead to the following puzzle.
Starting from the outside pentagon, move the
points along the boundary of the blue circle to change this into a
pentagon fully inside (so that it looks like a medallion with all the
points of the star on the blue circle). This is not easy! And when you're
done, turn it outside-in again to prove that it was not an accident...
Created with Cinderella by Leo Dorst 20010716.