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Looking at this page and trying to follow this step in the algebra: $2A(\Omega) = \sum_{i=0}^{n-1}(x_iy_{i+1} - x_{i+1}y_i) = \sum_{i=0}^{n-1}(x_i + x_{i+1})(y_{i+1} - y_i)$

I understand that the sum of the signed area of the parallelograms formed by the 2 vectors from the origin is twice the area of the polygon. What I don't understand is how the first summation equals the second summation. Expanding the binomial in the second summation gives extra terms $-x_iy_i$ and $x_{i+1}y_{i+1}$. Hope someone can assist.

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Those extra terms cancel each other. Since $x_iy_i = x_{k+1}y_{k+1}$ if $i=k+1 \text{mod} n$.