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I am currently working to understand a combinatorics problem. Within the proof they state that "Two circles can have at most one common chord ..." and I do not understand why this is true. I've included a screenshot from my textbook.

enter image description here

The book then goes on to finish the proof. Now I can follow everything before the last paragraph, but have no idea why two circles can have at most one common chord (last full sentence of last paragraph).

Question: Can you please provide some explanation for why two circles can have at most one common chord?

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  • $\begingroup$ Can you think of an example of two circles intersecting with say 2 common chords? That attempt should give you an answer. $\endgroup$ – Peaceful Oct 19 '16 at 10:13
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    $\begingroup$ Failure to dream up a counterexample is not normally considered a rigorous proof. $\endgroup$ – Michael Kay Oct 19 '16 at 14:22
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    $\begingroup$ No. But it can be an excellent start. $\endgroup$ – John Oct 19 '16 at 16:42
  • $\begingroup$ If two circles touch at a point, wouldn't a multitude of lines pass through that point and touch both circles? $\endgroup$ – Wossname Oct 19 '16 at 22:35
  • $\begingroup$ Two circles * can* have more than one common chord but then they are the same circle (and have infinite chords in common) $\endgroup$ – Francesco Oct 20 '16 at 4:01
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Two distinct circles will intersect in at most two points. If they intersect in two points, they have a single common chord, namely the line segment joining those two points. If they intersect in fewer than two points, they have no common chords.

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    $\begingroup$ Well, I think the OP was asking why they have at most two points in common... $\endgroup$ – Cave Johnson Oct 19 '16 at 3:48
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If two circles have two chords in common, then they must have at least three points in common.

A circle can be completely determined by three points, as there are three unknowns: the $x$ and $y$ coordinates of the circle's center, and the radius.

If you want something more concrete: if you have two points $(x_0,y_0)$ and $(x_1,y_1)$, then you know that the center of the circle must be equidistant from both; so, there is an infinite line of points to choose from for the center, and choosing the center completely determines the radius.

When you introduce a third point $(x_2,y_2)$, you can now note that the center of the circle must be equidistant from $(x_0,y_0)$ and $(x_1,y_1)$, and must also be equidistant from $(x_0,y_0)$ and $(x_2,y_2)$. This gives two lines, whose intersection (if such exists) must be the center. Assuming there is a good point for the center, the radius must be the common distance from this point to all three $(x_i,y_i)$ pairs.

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    $\begingroup$ I don't think the last point is complete. If the x & y of the circle's center are distinct unknowns, then three points actually have six knowns (the x & y of each). Thus one might expect that two points (four knowns) could uniquely identify a circle. As this is not the case, it doesn't immediately follow that bumping this to three points (six knowns) is sufficient to determine it. -- Contrast with a line segment, which has four unknowns (the x & y coordinates of the endpoints) but is determined either by two points (the end points) or can't be defined by an infinite number of internal points. $\endgroup$ – R.M. Oct 19 '16 at 15:05
  • $\begingroup$ A data point is, in this case, a relationship between an $x$ coordinate and a $y$ coordinate, not the coordinates themselves. The argument isn't perfect, but it is illustrative. $\endgroup$ – Nick Peterson Oct 19 '16 at 15:09
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    $\begingroup$ If you want something more concrete: if you have two points $(x_0,y_0)$ and $(x_1,y_1)$, then you know that the center of the circle must be equidistant from both; so, there is an infinite line of points to choose from for the center; radius is then determined. When you introduce a third point $(x_2,y_2)$, you can now note that the center of the circle must be equidistance from $(x_0,y_0)$ and $(x_1,y_1)$, and must also be equidistant from $(x_0,y_0)$ and $(x_2,y_2)$. This gives two lines, whose intersection (if such exists) must be the center, and you can immediately determine the radius. $\endgroup$ – Nick Peterson Oct 19 '16 at 15:10
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Two circles can have at most two common points (they can intersect at 2 points at most). Hence they can have only one common chord.I hope this image is illuminating.

Two circles with common chord

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A circle is defined by three non-collinear points. Two chords are defined by at least three points, so by taking two chords of a circle, you take at least three points, which uniquely defines your circle.

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