The problem: given vertices that define a polygon, determine whether another point $P$ lies inside the polygon or not.
Proposed solution: suppose we have a triangle $A, B, C$. Calculate the areas of the triangles that $P$ forms with the other points: $PAB, PBC, PAC$. Iff $P$ lies inside the triangle, then $\text{area}(PAB) + \text{area}(PBC) + \text{area}(PAC) = \text{area}(ABC)$.
I found this method described here, and it seems to work for any polygon. Is this the case? If so, is there a reference? I can't find it listed as any of the common solutions to this problem.
Sure; that works. But you have to be sure to take the area rather than the signed area, which may mean throwing in an absolute value. It's far more efficient to test that for each vertex U, your fourth point X lies on the same side of the edge not containing U as does U. So if U = A, then you test U against BC and X against BC, and if the signs match, you're good. Repeat for the other two edges.
If you have millions of points to test as your "fourth point" while the triangle remains constant, you can win big by doing some precomputation. Tomas Akenine-Moller has written many papers on this subject, and they're probably the best thing out there.