Position vertices to maximize the number of $3-$faces

Question

Position $n$ nodes on a plane such that the maximum number of $3-$faces is maximized in a way that the edges do not overlap and a node is not inside a chosen face.

Answer 1

The points and triangles in the arrangement with the maximal number of triangles must form a maximal planar graph: it is impossible to add more edges (i.e. create new triangles) without crossing edges. The following paragraph was directly taken from Wikipedia.

If a maximal planar graph has $v$ vertices with $v>2$, then it has precisely $3v-6$ edges and $2v-4$ faces.

One of those faces serves as the outer boundary of the arrangement, so we are left with at most $2n-5$ triangles as desired.