Consider an equilateral ABC with one of the vertex say A on the origin and side AB along X-axis. So coordinates of the vertices can be assigned as

A(0,0); B(a,0) and C(a/2, a3/2)

If masses m, 2m and 3m are placed on the vertices A, B and C respectively then coordinates of center of mass C(X,Y) can be expressed as

 

X = [0*m + a*2m + (a/2)*3m]/6m

 

or X= 7a/12

 

Similarly Y = [0*m + 0*2m + (a3/2)*3m]/6m = a3/4

 

so Y = a3/4

 

Now consider coordinates of square ABCD with side 'a' to be

A(0,0); B(0,a); C(a,a) and D(0,a) with masses m, 2m, 3m and 4m respectively and try to locate the center of mass.