Let be polygon with its ordered vertices (the vertices you take in order as you travel around it) be .

Also say they are taken in a counterclockwise fashion.

Let be the line segment from to for . But for define .

Define by .

Let denote the interior of the polygon, then by Green's theorem:

.

Thus, the area is equal to:

For define for . Notice that parametrizes .

Thus, .

Now, , so , while .

Therefore,

So if you add these up including you get:

Work on simplying that expression now.