If we used barycentric coordinates, wouldn't we be able to easily distinguish between all seven cases?

Does barycentric interpolation work for points outside the triangle?

I think barycentric coordinates won't work for distinguishing all these cases. It only calculates the distance ratio, but distance should always be nonnegative?

I a little bit confused about the term "possibility" here. What does "possibility of closest point" refer to?

Can you explain the "possibility" a bit more here?

Using barycentric interpolation, we can find the location relative to the triangle.

If a query point is inside the triangle, would the closest point not be itself?

If we use barycentric coordinates just to tell whether the point is in the triangle, would that make the operation cheaper?

What is possibility? Do you mean that all the possible options to place the point around the triangle ?

I get that people ask about barycentric coordinates, but aren't those specific to each triangle? So how would we use those coordinates for points outside the triangle?

Could we write these cases into one matrix?