What are "half-space tests" for?

I don't follow the wording on the slide, but I think this is the algorithm it describes:

• find distance from point to plane of triangle

• project point onto plane of triangle, check if projected point is in triangle:

  if so, closest point is the projection

  else, reduced to 2D case

I think half-space tests here means what we learnt in lecture 2 : http://cs248.stanford.edu/winter19/lecture/drawtriangle/slide_050