XXVII Olimpiada Matemática Rioplatense

Let $P$ be a point in the exterior of a circumference $\Gamma$, and let $PA$ be one of the tangents from $P$ to $\Gamma$. The line $l$ passes through $P$ and intersects $\Gamma$ in $B$ and $C$, with $B$ between $P$ and $C$. Let $D$ be the point symmetric to $B$ with respect to $P$. Let ${\omega }_1$ and ${\omega }_2$ be the circumferences circumscribed to the triangles $DAC$ and $PAB$ respectively; ${\omega }_1$ and ${\omega }_2$ intersect in $E\ne A$. The line $EB$ intersects ${\omega }_1$ in another point $F$. Prove that $CF=AB$.