Consider the following pair of circles
The above circles do not intersect each other at any point. Therefore, they do not have any point in common
The above circles touch each other only at one point Y. Therefore, there is 1 point in common
The above circles touch each other at 1 point X only. Therefore, the circles have 1 point in common
These circles intersect each other at two points G and H.
Therefore, the circles have two points in common.
It can be observed that there can be a maximum of 2 points in common.
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