Hello, chil2e!
Two distinct circles and a square lie in the plane.
What is the maximum number of points of intersection of 2 or more of these 3 figures? We can "talk" our way through most of it.
Let $\displaystyle S, C_1, C_2$ represent the square and two circles.
We can "see" that:
. . $\displaystyle S$ and $\displaystyle C_1$ intersect in at most 2 points.
. . $\displaystyle S$ and $\displaystyle C_2$ intersect in at most 2 points.
. . $\displaystyle C_1$ and $\displaystyle C_2$ intersect in at most 2 points.
The maximum number of intersections is: .$\displaystyle 2+2+2 \:=\:6$
Bottom line: is such a figure possible? Code:
* * * * * *
* * * *
* o *
* * * *
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* * - - o - o - - * *
* | * * | *
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* | * * | *
* | o | *
* | * * | *
* o * * o *
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* - - - - - - - *
Yes!