In the plane, consider a point
and a polygon
(which is not necessarily convex). Let
denote the perimeter of
, let
be the sum of the distances from the point
to the vertices of
, and let
be the sum of the distances from the point
to the sidelines of
. 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 $d^2 - h^2\geq\frac {p^2}{4}.$
Školjka