There are pesky edge conditions where a point is exactly on the common edge of two adjacent triangles. The point cannot be in both, or neither of the triangles. You need an arbitrary but consistent way of assigning the point. For example, draw a horizontal line through the point. If the line intersects with the other side of the triangle on the right, the point is treated as though it is inside the triangle. If the intersection is on the left, the point is outside.
If the line on which the point lies is horizontal, use above/below.
If the point is on the common vertex of multiple triangles, use the triangle with whose center the point forms the smallest angle.
More fun: three points can be in a straight line (zero degrees), for example (0,0) - (0,10) - (0,5). In a triangulating algorithm, the "ear" (0,10) must be lopped off, the "triangle" generated being the degenerate case of a straight line.