KöMaL problem A, 2017

Geometry Difficulty 6.0 National Olympiad Prove it Hungary

Let $A$ and $B$ be two vertices of a convex polygon $\mathcal{P}$ with maximum distance from each other. Let the perpendicular bisector of the segment $AB$ meet the boundary of $\mathcal{P}$ at points $C$ and $D$ . Show that the perimeter of $\mathcal{P}$ is less than $2(AB+CD)$ .

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