TR-MO 1997 Problem 5

In a triangle $ABC$ , the inner and outer bisectors of the $\angle A$ meet the line $BC$ at $D$ and $E$ , respectively. Let $d$ be a common tangent of the circumcircle $(O)$ of $\triangle ABC$ and the circle with diameter $DE$ and center $F$ . The projections of the tangency points onto $FO$ are denoted by $P$ and $Q$ , and the length of their common chord is denoted by $m$ . Prove that $PQ = m$