I have a rectangular piece of paper with a circle printed on it. I also have a handy-dandy writing utensil. How can I locate and mark the center of the circle?

![Picture](https://i.stack.imgur.com/xUivs.png)

Here's some technicalities:

* The paper is "infinitely thin," perfectly foldable, opaque, and a perfect rectangle. The circle is also perfect.
* The circle does not overlap, extend past, or touch the edges of the page.
* The paper can be any size rectangle and the circle can have any size radius, but these are out of our control.
* The edges of our folded page serve as our straightedge. We can also use the lines generated by the creases.
* One thing that we cannot do is simply guarantee well in advance that our folds occur at any particular angle. (I've found a really simple solution that involves this.)
* Our writing utensil is a tool of ridiculous precision (and accuracy).
* We don't have any access to third party services or tools.

Once the simplest version of this puzzle is solved, I am very interested in how we can change the puzzle and have it remain solvable.

* Can it be solved if the edges of the circle are allowed to touch the edges of the page?
* Can it be solved if we are not allowed to create tangents to the circle?
* Can it be solved if the paper is infinitely tall and wide, so that we only have access to one corner and two edges?

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This question is somewhat inspired by / based off of [a similar puzzle](https://math.stackexchange.com/questions/205953/how-do-you-find-the-center-of-a-circle-with-a-pencil-and-a-book) which allowed the use of book, but which did not specify a rectangular piece of paper.