In the acute isosceles triangle $ABC$ the altitudes $BB_1$ and $CC_1$ are drawn, which intersect at the point $H$. Let $L_1$ and $L_2$ be the feet of the angle bisectors of the triangles $B_1AC_1$ and $B_1HC_1$ drawn from vertices $A$ and $H$, respectively. The circumscribed circles of triangles $AHL_1$ and $AHL_2$ intersects the line $B_1C_1$ for the second time at points $P$ and $Q$, respectively. Prove that points $B, C, P$ and $Q$ lie on the same circle. (M. Plotnikov, D. Hilko)