I dont think theres an issue, the point P' simply belongs to both sides of itself. With this convention in place it is still true that there are two points in t

I dont think theres an issue, the point P' simply belongs to both sides of itself. With this convention in place it is still true that there are two points in the same side, and the proof goes through verbatim. Its mostly a question of whether you count the line defining a half-plane as belonging to the half plane or not, and clearly they do here