Sure, here is how you can extend the function to check if N points are coplanar, for any positive value of N:

```python
def are_n_points_coplanar(points):
  """
  Determines whether N points lie on the same plane.

  Args:
    points: A list of N points in a three-dimensional space.

  Returns:
    True if N points lie on the same plane, False otherwise.
  """

  # Check if any N - 1 of the points are collinear.
  for i in range(N - 1):
    for j in range(i + 1, N):
      if are_three_points_collinear(points[i], points[j], points[i + 1]):
        return True

  # If all N - 1 points are not collinear, then they must all lie on the same plane.
  return False
```

This function works by first checking if any N - 1 of the points are collinear. If they are, then it means that all N points must lie on the same plane. Otherwise, the function returns False.

The function are_three_points_collinear is the same as the one we defined earlier.

To check if N points are coplanar, you can call the are_n_points_coplanar() function with a list of N points as the argument. For example, to check if 5 points are coplanar, you would call the function as follows:

```python
are_n_points_coplanar([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12], [13, 14, 15]])
```

This function will return True if the 5 points lie on the same plane, and False otherwise.

I hope this helps! Let me know if you have any other questions.
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