Here, I've just used the unit normal vector to thicken the border of
the main triangle to distinguish it from the sub-triangles (thinner,
colored borders).

Even thicker borders, with bigger points for the vertices, also:

If we extend the sides of the triangle, we can make out how the
different regions are related to the way the subtriangles overlap.

If P is in a vertical angle [Is this the right term?] of one of the
vertices, then the exterior subtriangles won't overlap the interior
ones.

If we move P over into the "shadow" of the third side, BC, (imagining
point A to be emitting light beams in all directions), then the third
sub-triangle (yellow)overlaps the triangle, while the second
subtriangle (light blue) remains outside the triangle.

Or, just in terms of the subtriangles, subtriangles two and three
overlap.