<Tri-2024-2163>, hint

1>, Rotate triangle AMB around A  by 60 degrees

 such that M--->M'. B--->B' and B' M' M, C are collinear

Hence  min(AM+BM+CM)=B'C.

2> Let O be the intersection of AC with BD and M'BA is equilateral triangle

similarly N'DC is also equilateral. Since OAB and OCD are congruent AOC are collinear

M'N' is the required one.