# How do I find the center of a circle on a rectangular piece of paper?

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?

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?

This question is somewhat inspired by / based off of a similar puzzle which allowed the use of book, but which did not specify a rectangular piece of paper.

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What are the conditions of the paper?density, type, these things? – HipsterMathematician Oct 15 '12 at 1:26

1. Fold a crease crossing your circle. This creates an edge
2. Keep the first crease and fold a second crease passing through an intersection of the previous crease with the circle and perpendicular to the crease of step 1. This crease splits the edge of step 1 in two. The perpendicularity is done by ensuring that the two edges appearing from the edge of step 1 superimpose.
3. Using the same process as in step 2 fold a rectangle inside your circle by folding two more times and using the intersection points between the previous creases and the circle.
4. Fold the diagonals of this rectangle, you get the center at their intersection. You can actually save the last edge of the rectangle in step 3 since you already have the 4 vertices of the rectangle at this point.

I believe this solution doesn't require to be able to look through the paper by transparency (otherwise folding a diameter of the circle is straightforward), and doesn't use any edge or corner of the paper. No tangents are used.

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This construction uses the fact that perpendicular bisector of a chord goes through the center of the circle.[See picture]

First, fold the paper so the crease forms a chord across the circle. Now, crease the paper at the points where the chord intersects the circle. Then, fold the paper perpendicular to the chord, so the two creases overlap. Unfold, and you should see that the new crease is the perpendicular bisector to the chord, and thus goes through the center of the circle.

Repeat with another chord. The intersection of the two perpendicular bisectors will be the origin.

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"Then, fold the paper perpendicular to the chord, so the two side of the circle overlap to form a semicircle." It's not clear to me how you can fold the paper such that the two halves of the circle overlap. This would be easy if the paper were transparent, but it's not. – Tanner Swett Oct 15 '12 at 3:51