We have
Then,
Your notation is slightly confusing. You say that so "AB" means the vector from A to B but then you use BM and MC as lengths of vector since you are dividing them. I don't know whether the "AM" you want is to be a vector or a length.
I am going to call the vector from B to C, [tex]\vec{c}[/itex], the vector from B to M, , and the vector from A to M, and use the capital letters only for the lengths.
Since , BM= kMC and BM+ MC= (k+1)MC.
Then . But so and then so, finally,
The distance AM is the length of the vector above.