KöMaL 2022 B5257

In an acute-angled triangle $ABC$, the heights are $AA_1$, $BB_1$, $CC_1$, and the midpoint of side $AB$ is $F$. A circle $k$ passes through the points $F$ and $C_1$, and intersects the extensions of line segments $A_1C_1$ and $B_1C_1$ beyond $C_1$ at points $P$ and $Q$, respectively. Prove that $A_1P=B_1Q$.