In the plane, consider a point
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
and a polygon
![\mathcal{F}](/media/m/7/8/2/7827734528ed1d36d9902b6cde16021c.png)
(which is not necessarily convex). Let
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
denote the perimeter of
![\mathcal{F}](/media/m/7/8/2/7827734528ed1d36d9902b6cde16021c.png)
, let
![d](/media/m/f/7/d/f7d3dcc684965febe6006946a72e0cd3.png)
be the sum of the distances from the point
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
to the vertices of
![\mathcal{F}](/media/m/7/8/2/7827734528ed1d36d9902b6cde16021c.png)
, and let
![h](/media/m/e/4/3/e438ac862510e579cf5cbdbe5904d4ba.png)
be the sum of the distances from the point
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
to the sidelines of
![\mathcal{F}](/media/m/7/8/2/7827734528ed1d36d9902b6cde16021c.png)
. Prove that
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In the plane, consider a point $X$ and a polygon $\mathcal{F}$ (which is not necessarily convex). Let $p$ denote the perimeter of $\mathcal{F}$, let $d$ be the sum of the distances from the point $X$ to the vertices of $\mathcal{F}$, and let $h$ be the sum of the distances from the point $X$ to the sidelines of $\mathcal{F}$. Prove that $d^2 - h^2\geq\frac {p^2}{4}.$