Question 129
Let $\omega_{1}$ and $\omega_{2}$ be two circles that intersect in $P$ and $Q$. It is assumed that the tangent common to $\omega_{1}$ and $\omega_{2}$, the closest to $P$, affects circles $\omega_{1}$ and $\omega_{2}$ In $A$ and $B$ respectively. The tangents in $P$ to circles $\omega_{1}$ and $\omega_{2}$ intersect $\omega_{2}$ and $\omega_{1}$ in $D$ and $C$ respectively. $H$ and $K$ are placed on the segments $[A C]$ and $[B D]$ so that $A H = A P$ and $B K = B P$. Show that points $A, B, K, Q$ and $H$ are on the same circle. 