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\begin{align}
S&=\frac{1}{2}\left|\sum_{k=1}^{n}(x_{k}y_{k+1}-x_{k+1}y_{k})\right|\\
&=\frac{1}{2}\left|\sum_{k=1}^{n}(x_{k}-x_{k+1})(y_{k}+y_{k+1})\right|\\
&=\frac{1}{2}\left|\sum_{k=1}^{n}x_k(y_{k-1}-y_{k+1})\right|\\
&=\frac{1}{2}\left|\sum_{k=1}^{n}y_k(x_{k-1}-x_{k+1})\right|
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