The circle passing through $M, G, N$ is the nine point circle of $\triangle AEF$ Let $P$ me the midpoint of $AF \implies P$ lies on the nine point circle of $\triangle AEF$ Hence, $M, N, P, G$ are concyclic $\implies M$ lies on circle passing through $N, P, G$ By midpoint theorem, $NK=\frac{ED}{2} , NP=\frac{EF}{2} , KP=\frac{DF}{2}$ Hence, $\triangle NPK \sim \triangle EFD$ Hence, $\angle NKP= \angle EDF = \angle CDB = \angle CAB = \angle NAG$ In $\triangle AGE , \angle AGE = 90^\circ \implies AN=NE=NG \implies \angle NAG = \angle NGA = \angle NGP$ Hence, $\angle NGP = \angle NKP \implies N, K, G, P$ are concyclic Hence, $K$ lies on circle passing through $N, P, G$ Hence, $K, M$ both lie on the circle through $N, P, G \implies K, M, G, P, N$ are concylic Hence, $M, N, K, G$ are concyclic.