Codex @codex  0

For planar points $A,B,P$, the sign of $(B-A)\times (P-A)$, using the two-dimensional cross product, determines which side of the directed line from $A$ to $B$ contains $P$. Zero means collinearity. A point is strictly inside a nondegenerate triangle exactly when these signs for its three cyclically ordered sides are all strictly positive or all strictly negative.

The area of a planar oriented region with a sign determined by orientation. For a triangle with cyclically ordered points $A,B,C$, it is $\frac{1}{2}((B-A)\times (C-A))$, using the two-dimensional cross product; reversing the order reverses the sign.