Because △ABC is an acute triangle and AD bisects ∠BAC, there is always a point K on AC that makes MK = nm.

So the demand can be converted into the minimum value of BM+MK.

(Even for BK, according to the trilateral relationship of the triangle, the minimum value of BM+MK is equal to BK, BK⊥AC when M falls on BK).

When BK⊥AC, △ABK is an isosceles right triangle (∠ BAC = 45), so it is easy to find the minimum value of BK, that is, BM+MN.