Two points determine a unique line. Therefore, there are infinitely many circles that can pass through two given points. This is because a circle can be defined by its center, which can lie anywhere along the perpendicular bisector of the line segment connecting the two points.
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It takes 3 non collinear points to define one specific circle. With only two points an infinite number of circles can be drawn.
Proof:
Given two points A, B draw the line between them. Then find the perpendicular bisector of the line AB. Any point on the perpendicular bisector is equidistant from the two original points, A and B. A circle with center C and radius AC will then pass through points A and B. There are infinite point C's on the perpendicular bisector so there are infinite circles.
Given three points A, B and D you can find the perpendicular bisector for line segements AB and then the perpendicular bisector fof line segment BC. The two perpedicular bisectors will not be parallel because the points A, B and D are non collinear. This means the two perpeniducar bisectors will intercept at only one point C(like any two intercepting lines). This point C is equidistant from points A, B, and D. A circle with center C and radius AC will then pass through all three of the points. Since there is only one point C that lies on both perpendicular bisectors, there is only one circle possible.
infinite.. you can have different sizes of circles crossing at the same point.. so it can literally be an infinite amount
There are an infinite number of planes that pass through a pair of points. Select any plane that passes through both the points and then rotate it along the line joining the two points.
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Nonee !
A line, ray, or line segment contains an infinite number of points.