Ayanna Trujillo

2022-06-25

P is an arbitary point , and , where AM is a median. Observe that $x=\frac{2}{3}$ when $p=\frac{1}{2}$ . Find a general formula for x in terms of p.

Cristian Hamilton

Expert

Step 1
If 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 $AX:XM=m:2$ ;
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 $AP:PC=m:1$ .
From these two relations you can eliminate m and find x as a function of p.

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