I need to find the center of a circle using compass and ruler (or preferably just a straightedge) without using the interior of the circle. In other words, no line or point can be used which is inside the circle. All lines and points used in the construction must be outside the circle.
The center can be defined by points outside the circle. For example, you could identify 4 points outside the circle. The intersection of the lines going through each pair of points indicates the center.
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Choose any two points on the circle. Set the radius of your compasses larger than the diameter of the circle, and draw a circle centered on each point. These circles intersect in two points that form a line through the required center.
If the compasses are too small, or collapse when you lift them off the paper, construct a hexagonal grid of points until you have two points diametrically opposite each other.
Choose any two points on the edge of your circle. Connect these two points with a line segment, this forms a chord in your circle.
You can show that the perpendicular bisector of any chord of a circle goes through the center of the circle.
Therefore, to find the circle center, construct two different chords of the circle, and find the perpendicular bisector of each chord. The circle center is where these bisectors meet.