The area of a closed polygonal plane curve can be defined using the shoelace formula. Let $A_1$,..., $A_n$ be the vertices of the curve and $O$ be some origin in the plane. Set $A_{n+1} = A_1$.
$$ \mbox{area}(A_1,..., A_n) = {1 \over 2}\sum_{i=1}^n \det(\overrightarrow{OA_i}, \overrightarrow{OA_{i+1}}) $$
I am looking for an elementary (algebraic) proof that a polygon has a non-zero area. Equivalently, for a proof that a polygonal curve with zero area has a self-intersection.
By elementary, I mean I don't want to rely on the existence of triangulations for polygons nor on the Jordan separation theorem since I would like to use the shoelace formula to prove these results.
Use Gauss or Greens's theorems to locate a point of self intersection for zero net area.