KöMaL 2023 B5333

The foot of the altitude drawn from vertex $A$ of an acute-angled triangle $ABC$ is $T_A$. The ray drawn from vertex $A$ through the centre $O$ of the circumcircle intersects side $BC$ at point $R_A$. Denote the midpoint of line segment $AR_A$ by $F_A$. Starting from vertices $B$ and $C$, obtain the points $T_B$, $R_B$, $F_B$, $T_C$, $R_C$, $F_C$ in the same way. Show that the lines $T_AF_A$, $T_BF_B$ and $T_CF_C$ are concurrent.