Thought you might be interested in this as an example of illustrating
algebraic geometry with Graphing Calculator. Free GC viewers are
available at http://www.pacifict.com/FreeStuff.html

Simplified the coloring routines. My previous example was needlessly
complicated. Still painfully slow on my iMac, unfortunately. But good
for making illustrations, perhaps, more than for interactive use,
unless you've got one of the latest computers.

Here we can see better how the "code" works. Each component of the
vector returned by T(A, B, C, P) represents the area of the
sub-triangle formed by the test point P and two of the vertices of the
triangle. That is,

is the area of the first sub-triangle, triangle PAB

is the area of the second sub-triangle, triangle PBC

is the area of the third sub-triangle, triangle PCA

So, as the second sub-triangle "takes over" the whole area of the
triangle, that means that the test point must be approaching the first
vertex.

If the third sub-triangle is the biggest, that means we're near the
second vertex (the "previous" one if the vertices are considered in
cyclic order).

If the first sub-triangle is the biggest, that means we're near the
third vertex (the "previous" one if the vertices are considered in
cyclic order).

If sub-triangle ABP is approaching zero while the other two
sub-triangles are positive, that means the test point P is near side
AB:

If we go across side AB, the triangle APB becomes negative in area,
This is shown by a green hue, which is unfortunately covered up by the
other triangles. I need to do some kind of cross-hatching here.

If we "flip the triangle over", by making the points in clockwise
order, than the area of the triangle as a whole becomes negative.

Here's an attempt to have striped coloring. Can't get negative areas
yet, and it's too slow to be usable on my iMac.
Cross-hatching with lines might be OK, though.

If it worked, it would show how the areas of the sub-triangles with
respect to an external point add up to the area of the triangle ABC,
since the part that's outside is negative in area! See Klein,
"Elementary Mathematics from an Advanced Standpoint: Geometry", page 7
for details.