Delaunay triangulation

Triangulation of a set of points on the plane is a set of nondegenerate triangles that satisfy the following conditions:

1. The vertices of the triangles are only the points of the original set. Each point of the initial set is the vertex of at least one triangle.

2. Any two different triangles either have no points in common, share a common vertex, or share a common side (but the area of their intersection is always $0$).

3. Any point located inside the convex hull of the original set of points belongs to at least one triangle (it may belong to several triangles, if it is a common vertex or lies on a common side). (The convex hull of a set of points is the smallest convex polygon containing all the points).

A triangulation is called a Delaunay triangulation if, in addition, the following condition is satisfied:

• The interior of the circle circumscribed around any triangle of the triangulation does not contain any of the original points (points may lie on the circle; in particular, obviously, the vertices of the triangle lie on the circle).

For the given set of points, determine the number of its Delaunay triangulations (two triangulations are considered different if they differ by at least one triangle).