Using each edge of the hexagon as one of the bases of a triangle, only six triangles can be inscribed within a regular hexagon. A new vertex should be added at the midpoint of the hexagon to create the remaining triangles, which will share this new vertex.
Technically, someone could add an infinite number of triangles to the inside of a hexagon. Drawing a single point in the middle of the hexagon is only one way to achieve inscription. It is possible to then bisect all of those triangles again to produce 12 triangles, or once more to produce 24 triangles.