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Therefore, to find a point that is equidistant from three points, draw the perpendicular bisectors of the lines joining two pairs of these points. Their point of intersection is equidistant from all three points (provided, of course, that they are not parallel).
This point is, of course, the centre of the circle passing through the three given points, and is always a unique point provided that the three points don't lie in a straight line. But in general, you won't be able to do this for four (or more) points, unless they happen to form a cyclic quadrilateral.