In an acute-angled triangle $ABC$, the altitudes from $A,B,C$ meet the sides of $ABC$ at $A_1$, $B_1$, $C_1$, and meet the circumcircle of $ABC$ at $A_2$, $B_2$, $C_2$, respectively. Line $A_1 C_1$ intersects the circumcircles of triangles $AC_1 C_2$ and $CA_1 A_2$ at points $P$ and $Q$ ($Q\neq A_1$, $P\neq C_1$). Prove that the circle $PQB_1$ touches the line $AC$.