MathNet 2022-part-01 — 197 problems (EN–VI) Question 177

As illustrated in Fig. 1.1, $ABCDEF$ is a cyclic hexagon. The extensions of $AB$ and $DC$ meet at $G$; the extensions of $AF$ and $DE$ meet at $H$. Let $M$ and $N$ be the circumcentres of $\triangle BCG$ and $\triangle EFH$, respectively. Prove that the lines $BE$, $CF$, and $MN$ are concurrent.


Fig. 1.1