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In General orchard planting problem for circles , the problem of 4 points per circle has been studied.

The problem here is that what is the maximum number of 5-point circles for a configuration of n points drawn on a plane?

It is easy to show that we need 8 points to get two 5-point circles and 9 points to get three 5-point circles enter image description here

10 points could reach five 5-point circles: enter image description here

11 points to reach seven 5-point circles and 12 points to reach nine 5-point circles: enter image description here

In all pictures above, one point is at infinite point and circle-inversion transformation (turn line into circle) could be used to transform it to normal point.

bobble
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Zhaohui Du
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2 Answers2

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Here is the 12 points, 12 5-circles solution I mentioned in a comment.

enter image description here

And adding a single point in the center gives you 3 more circles.

Florian F
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Add an equivalent result of Florian F. (one infinity point) for 13 points with 15 5-point circles (where black point and black dash circles are the extra point and 3 extra 5-point circles) enter image description here And similarly we could add one more extra point and 3 extra 5-point circles to reach 14 points with 18 5-point circles and one more point and 2 extra 5-point circles to reach 15 points with 20 5-point circles.

Zhaohui Du
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