CZ-SK-MO 2005 Problem 3

In a trapezoid $ABCD$ with $AB // CD, E$ is the midpoint of $BC$ . Prove that if the quadrilaterals $ABED$ and $AECD$ are tangent, then the sides $a = AB, b = BC, c =CD, d = DA$ of the trapezoid satisfy the equalities $a+c = \frac{b}{3} +d$ and $\frac1a +\frac1c = \frac3b$ .