About four months ago, I came up with several generalizations (intuitions) for Sylvester-Gallay's theorem, but it seems that as with the original Sylvester-Gallay theorem, it is difficult to interfere with these generalizations and prove them with simple Euclidean methods. I had some unsuccessful attempts to prove the circular generalization, please who can prove Or refute any of these generalizations, please be kind
- A three-dimensional generalization of Sylvester-Gallay's theorem: For any finite set of points in Euclidean space, there must be a plane that passes through exactly three of them, or a plane that passes through all of them.
- Circular generalization of Sylvester-Gallay's theorem: For any finite set of points in the Euclidean plane, there must be a circle that passes through exactly three of them, or a circle that passes through all of them
- A conic generalization of Sylvester-Gallay's theorem: For any finite set of points in the Euclidean plane, there is inevitably a conic section that passes through exactly five of them, or a conic section that passes through all of them.
- A spherical generalization of Sylvester-Gallay's theorem: For any finite set of points in Euclidean space, there must be a ball that passes through exactly four of them, or a ball that passes through all of them.
What I came up with might be useful in proving a circular generalization: If we have a point located inside a circle containing four points or more on its circumference, then inevitably there is a circle that passes through two of these points and passes through the inner point so that it is smaller than the basic circle (there is also a case of equality that occurs only if the four points form the vertices of a square)