KöMaL 2017 A693

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 $2(AB+CD)$.