OMCC 2014 Problem 5

Points $A$ , $B$ , $C$ and $D$ are chosen on a line in that order, with $AB$ and $CD$ greater than $BC$ . Equilateral triangles $APB$ , $BCQ$ and $CDR$ are constructed so that $P$ , $Q$ and $R$ are on the same side with respect to $AD$ . If $\angle PQR=120^\circ$ , show that

$$\frac{1}{AB}+\frac{1}{CD}=\frac{1}{BC}.$$