Question 97
Let $H$ be the orthocenter of an acute triangle $ABC$, and let $A_1, B_1, C_1$ be the feet of the altitudes belonging to the vertices $A, B, C$, respectively. Let $K$ be a point on the smaller $AB_1$ arc of the circle with diameter $AB$ satisfying the condition $\angle HKB = \angle C_1KB$. Let $M$ be the point of intersection of the line segment $AA_1$ and the circle with center $C$ and radius $CL$ where $KB \cap CC_1 = \{L\}$. Let $P$ and $Q$ be the points of intersection of the line $CC_1$ and the circle with center $B$ and radius $BM$. Show that $A, K, P, Q$ are concyclic.