I think this is finished, but would need checking.
If P is above any line coming from the uppermost point of the triangle, it is outside of the face. Likewise, if it is below a line from the lowest point, it is outside. If it is neither of these things, it MAY be in the face.
If P is either above or below BOTH lines coming from the leftmost point, it is outside. If P is above or below both lines from the rightmost point, it is outside. Otherwise, if MAY be in the face.
If both these say that the point MAY be in the face, it IS in the face. However, if there are horizontal or vertical lines in the triangle, the following checks must be performed. In the case where there is only either a horizontal or only a vertical line, the other check may be ommited.
If there is a horizontal line AB and point O, then if O is above AB, P is outside if P is below AB.
If O is below AB, then P is out when above AB. Otherwise, it MAY be in the face.
If there is a vertical line AB in the triangle, and the remaining point O is to the left of it, P will be outside the triangle if P is to the right of the line.
If O is to the right of AB, then P is ouside if it is left of AB. Otherwise, it MAY be in the face.