What is the maximal number of triangles in a planar graph with $n$ points?

Here by *triangle* I mean any closed path of length 3 (so a triangle can also include in the interior points other than the vertices).

For example, it is easy to see that the answer is 4 for $n=4$ and 7 for $n=5$.

VERY IMPORTANT EDIT: At the beggining, the question was for coplanar graphs. But I changed it to planar graphs now.