ZA-SAMO 1995 Problem 4

Three circles, with radii $p$ , $q$ and $r$ and centres $A$ , $B$ and $C$ respectively, touch one another externally at points $D$ , $E$ and $F$ . Prove that the ratio of the areas of $\triangle DEF$ and $\triangle ABC$ equals $$\frac{2pqr}{(p+q)(q+r)(r+p)}.$$