KöMaL problem A, 2021

In acute triangle $ABC$ the feet of the altitudes are $A_1$, $B_1$ and $C_1$ (with the usual notations on sides $BC$, $CA$ and $AB$, respectively). The circumcircles of triangles $AB_1C_1$ and $BC_1A_1$ intersect the circumcircle of triangle $ABC$ at points $P\ne A$ and $Q\ne B$, respectively. Prove that lines $AQ$, $BP$ and the Euler line of triangle $ABC$ are either concurrent or parallel to each other.