<Qua-2025-793>, hint

construct a triangle ADM' , which is congruent with triangle BCM, and  MN cuts AD at N' 

Then <ANN'=<MCB, <DNN'=<MBC. Since <AND+<AM'D=180, A,N,D,M are concyclic

<M'AD=<N'ND, N,N' M' are collinear. Hence M,N,N',M' are collinear and parallel to AB. 

MN//AB