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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.

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  • $\begingroup$ What's the red circle in your second picture point to? $\endgroup$
    – justhalf
    Commented Jan 26, 2023 at 2:51
  • $\begingroup$ great idea for a problem $\endgroup$
    – Dmitry Kamenetsky
    Commented Jan 26, 2023 at 4:14
  • $\begingroup$ @justhalf Ignore it. It shows that we could not add a third circle to the picture. $\endgroup$
    – Zhaohui Du
    Commented Jan 26, 2023 at 14:16
  • $\begingroup$ With 12 points you can make 12 5-point circles. Make a stereographic projection of an icosahedron. Or project a dodecahedron and you get 24 circles using 20 points. $\endgroup$
    – Florian F
    Commented Jan 26, 2023 at 23:29
  • $\begingroup$ astonishing result. Searching by computer and we could not find any combination with 13 5-point circles so that 12 5-point circles are best result of 12 points $\endgroup$
    – Zhaohui Du
    Commented Jan 27, 2023 at 3:38

2 Answers 2

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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.

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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.

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