P is an arbitary point , and p = A P A C . x = A X A M , where AM is a median. Observe that x = 2 3 when p = 1 2 . Find a general formula for x in terms of p.

Question

P is an arbitary point , and   p  =            A      P              A      C        .     x  =            A      X              A      M       , where AM is a median. Observe that   x  =      2    3   when   p  =      1    2   . Find a general formula for x in terms of p.

Cristian Hamilton · Accepted Answer

Step 1If you put two unit masses at B and C and a mass m at A, then you can find the center of mass in two different ways:1) put a mass of 2 at M (which is the center of mass of B and C) and then find X as the center of mass between M and A, so that   A  X  :  X  M  =  m  :  2 ;2) put a mass   m  +  1 at the center of mass of A and C and then find X as the center of mass between P and B; this can work only if the center of mass of A and C is at point P, so that   A  P  :  P  C  =  m  :  1 .From these two relations you can eliminate m and find x as a function of p.