Barycenter Command

This command differs among variants of English:

Barycenter (US)
Barycentre (UK + Aus)

Barycenter( <List of Points>, <List of Weights> ): Set the center of a system of points in the list, defined as the average of their positions, weighted by their value, using the proper formula.

Barycenter({(2, 0), (0, 2), (-2, 0), (0, -2)}, {1, 1, 1, 1}) yields point A(0, 0)
Barycenter({(2, 0), (0, 2), (-2, 0), (0, -2)}, {2, 1, 1, 1}) yields point B(0.4, 0). The x-coordinate of this point was determined by $\frac{1}{ 2+1+1+1 }*(2*2+1*0+1*(-2)+1*0)$ = $\frac{1}{ 5 }*2$ = 0.4