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Polygon F and d^2 - h^2 >= p^2 /4

Source: IMO Shortlist 1996, G9

August 9, 2008

geometryperimeterinequalitiesPythagorean Theoremgeometric inequalityIMO Shortlist

Problem Statement

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 \minus{} h^2\geq\frac {p^2}{4}.

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