<Tri-2024-2162>, hint

Suppose M is the incenter  I of ABC, then BA'+CB'+AC' is constant 

 Suppose M' in the AC'IB. and orthogonal projection of M' onto BC, CA, AB respectively

be A",B",C". Then A'A"+B'B"-C'C" >0. Because A'A", B'B", C'C" are the sides of a triangle.

Hence the locus of M is the incenter I of triangle ABC.