Hi

A (13,13) B (9,3) and C (-1,-1)

N is the midpoint of [AC] so

coordinates are

is a direction vector of the line (BN), so is whose coordinates are (-1,1)

For a general Cartesian equation ax+by+c=0, we know that one direction vector has coordinates (-b,a)

Here -b=-1 and a=1

A Cartesian equation of (BN) is therefore x+y+c=0

The value of c is found by substituting B (or N) coordinates into the equation : c=-12

x+y-12=0

Your equation of (AM) is right : 4x-3y=13

P is the point of intersection of AM and BN therefore

Which gives P(7,5)

M(4,1) and P(7,5) (3,4)

and

M(4,1) and A(13,13)

and

Therefore MP = MA / 3