Consider a convex $n-$gon in $\mathbb{R}^2$ with sides contained in the lines $y=k\_ix+b\_i, 1\leq i\leq n.$ Then its area equals to
$$
S=\frac{1}{2}\sum\_{i=1}^{n} \frac{(b\_{i+1}-b\_i)^2}{k\_{i+1}-k\_i}.
$$
Of course, it is not hard to prove this formula, but it still looks a bit mysterious to me. 


**Question** Has this expression appeared somewhere? It will be wonderful to see a pretty proof of it.


**Remark** Another expression for $S$ can be found [here](https://arxiv.org/pdf/1012.6030.pdf), formula (17). It is a toy analog of a MHV scattering amplitude in N=4 SYM. 



Strange formula for area of a convex polygon
Denote $A\_i=(0,b\_i)$. It is a point on $\ell\_i=\{(x,y):y=k\_ix+b\_i\}$, and let $$P\_i=\ell\_i\cap \ell\_{i+1}=\left(\frac{b\_{i+1}-b\_i}{k\_i-k\_{i+1}},\frac{k\_ib\_{i+1}-k\_{i+1}b\_i}{k\_i-k\_{i+1}}\right)$$ be the vertex of the polygon. The (properly oriented) area of $\triangle A\_iP\_iA\_{i+1}$ equals $\frac{(b\_i-b\_{i+1})^2}{2(k\_{i+1}-k\_{i})}$, and the sum of such oriented areas is the oriented area of the polygon.


