KZ-MO 2001 Problem 7

Two circles $w_1$ and $w_2$ intersect at two points $P$ and $Q$ . The common tangent to $w_1$ and $w_2$ , which is closer to the point $P$ than to $Q$ , touches these circles at $A$ and $B$ , respectively. The tangent to $w_1$ at the point $P$ intersects $w_2$ at the point $E$ (different from $P$ ), and the tangent to $w_2$ at the point $P$ intersects $w_1$ at $F$ (different from $P$ ). Let $H$ and $K$ be points on the rays $AF$ and $BE$ , respectively, such that $AH = AP$ and $BK = BP$ . Prove that the points $A$ , $H$ , $Q$ , $K$ and $B$ lie on the same circle.