Two Circles Can Have At Most One Common Chord? (IMO) 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.

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? 
 A: 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.
A: 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.
A: 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.
A: 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.

