International Mathematical Olympiad 1989 Problem 4

Let $ABCD$ be a convex quadrilateral such that the sides $AB$, $AD$, $BC$ satisfy $AB = AD + BC$. There exists a point $P$ inside the quadrilateral at a distance $h$ from the line $CD$ such that $AP = h + AD$ and $BP = h + BC$. Show that:

$$\frac{1}{\sqrt{h}} \geq \frac{1}{\sqrt{AD}} + \frac{1}{\sqrt{BC}}.$$