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  1. Several points on a plain called "initial points" are given, which is the point set at first.
  2. Make circles by every three points in the point set that haven't been connected and don't lie on the same line.
  3. Add the intersections into the point set.
  4. Go to STEP 2.

Note that three points may lie on the same line, or/and four points may lie on the same circle. The initial points have "infinite intersection" property iff new intersections can be made infinitely. What kind of initial points have the property?

Example (Words aren't important): enter image description here

Most initial points have the property, but here are some examples that don't:

  1. Any $1/2/3/4$ point(s).
  2. At least $5$ points on the same line.
  3. At least $5$ points on the same circle.
  4. Vertexes and center of a square.

Your task is to classify the cases that don't have "infinite intersection" property.

Background. I've read the original question in a book, which the only difference is:

  1. Connect every pair of points in the point set that haven't been connected.

Most initial points have the property, except the following three cases:

  1. Any $1/2/3/4$ point(s).
  2. At least $3$ points on the same line + $2$ points.
  3. At least $4$ points on the same line + $1$ point.
  4. At least $5$ points on the same line.
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