How to square a circle

A few days ago, I wrote a post in which I introduced a simple and elegant method to "square" a circle using only compass and straightedge. The solution in that post gave a value for that was accurate to 7 decimal places.
One of the readers left a comment asking for a way to get a better approximation for , using only compass and straightedge.

While it is impossible to square a circle, because is transcendental, there exists a simple algorithm to get a construction for any desired approximation. In this post I will not describe a specific construction, but a general algorithm for finding a construction, given an error you want to get.

Lets look on the following circle:


This is a unit circle, with the point marked in red. Our goal is to get a construction that will give us a point B on the radius so that the distance between B and the red dot is as small as we want.
Point C is totally random - it can be any point on the circle or even outside of the circle. I put it only for convenience.

Step one : Connect C with O, the center of the circle.
Step two: The red dot is to the right of O, so mark a dot at the center of OD. Connect this new dot with C.
Step three: The red dot is still to the right of , so mark a dot at the center of D. Connect it with C.
Step four: Now the red dot is between and . Again mark the dot at the center of this segment. Connect it with C.
At this point the length of the line A is close to . If this is not good enough continue in the some way. With each step you will be getting closer and closer to the red dot.
You should get something like this:
It is very easy to see on this picture that each step brings you closer, and it happens relatively fast.
After you will find the sequence of steps that bring you close enough to the red dot, simply repeat it without marking the dot first.

Lastly, an exercise to the reader: prove that is impossible to square the circle this way.