The Sylvester–Gallai theorem in geometry states that, given a finite number of points in the Euclidean plane, either

All the points are collinear; or

There is a line which contains exactly two of the points.

I think the theorem is also true for circle, plane version and curve version, can you give a proof of these versions?

**Conjecture: (circle version):**

Given a finite number of points in the Euclidean plane, either

All the points are concyclic; or

There is a circle which contains exactly three points.

**Conjecture: (Plane version)**

Given a finite number of points in the three-dimensional space, either

All the points in a plane; or

There is a plane which contains exactly three points.

**Conjecture: (Curve version)**

Let $n, N$ be the natural number with $(\frac{3n+n^2}{2} < N < \infty)$, give $N$ points in the Euclidean plane, either

All the points lie on a curve of degree $n$; or

There is a curve of degree $n$ which contains exactly $\frac{3n+n^2}{2}$ of the points.