Figuring out if you're in one of the unambiguous regions (1, 2, 3 below), is fairly easy: treat each edge of your triangle as a 2D plane and test which side of the plane your new point is on. If you're inside two of them but outside one, then that one corresponds to the edge of the triangle that contributes two vertices to your new triangle.
If you're inside one and outside two, you're in the ambiguous case where the nearest part of the triangle to your new point is a corner. In that case, you can form a 2D plane from the midpoint of the opposite edge (the one that you're inside of) and the nearest vertex (the one shared by the two planes you're outside of). You can pick an edge depending on which side of this plane your new point is on.
Note that a plane test in 2D works the same way as in 3D: dot a vector from anywhere on the plane to your point with the plane's normal (in 2D, this is the line's perpendicular).
(Incidentally, the magenta-delimited regions in this image are called Voronoi regions; they're the areas of space containing points that are closest to a particular feature—edge or vertex—of the triangle. Edit: My terminology here isn't actually quite correct, these aren't exactly Voronoi regions.)