I was looking for a way to prove the shoelace formula when I found this proof:
For this clockwise order to make sense, you need a point O inside the polygon so that the angles form $OA_{i}A_{i+1}$ and $OA_{n}A_{1}$ are all positive.
Then the formula is just adding up the areas of the triangles $\Delta OA_{i}A_{i+1}$ and $\Delta OA_{n}A_{1}$.
So all you need is area of $\Delta OA_{1}A_{2}\; =\frac{\; b_{2}a_{1}\; -\; b_{1}a_{2}}{2}$, which is elementary.
However, I have several questions on this proof.
First, what does it mean by "so that the angles are all positive"? Could someone provide a diagram or explain what it means by this?
Second, how do you know such a point always exists?
Third, how did you get the formula $\Delta OA_{1}A_{2}\; =\frac{\; b_{2}a_{1}\; -\; b_{1}a_{2}}{2}$? I don't see how it's "elementary"? Could someone provide a proof on how the area of the triangle is that? Also, why doesn't the area depend on point $O$?