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A circle is the locus of points which are equidistant from a point (the center) pretty much defines the pythagoras circle equation x^2 + y^2 = r^2
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The locus of points which are equidistant from any two points is the perpendicular bisector of the line joining the points. This can be justified using greek construction so using a compass and draw arcs from each point and join the intersection points. Or note that each point on the locus is equidistant from the pair of points by drawing an isosceles triangle or ....
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So if you have 3 points lying on the circumference, picking two will give a chord and the perpendicular bisector must pass through the center as the centre is equidistant from every point on the circumference. The perpendicular bisector line is the locus of all possible circle centres for those two points.
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Drawing 2 chords will give two intersecting diameters which defines the center as all 3 points are equidistant from the intersection point/center. Obviously, by definition/construction/... the perpendicular bisector of the 3rd chord (pair of points) would pass through this point.