Definition of the barycenter
Take a triangular sheet and put on it, at point A, a 1 kg weight, at point B, a 2kg weight, and at point C a 3kg weight. The barycenter of the system is the centre of gravity of the sheet, that means the point where the sheet is balanced. Call it G. You guess that G will be closer to C than to A. G verifies the vector equality :
In general, the barycenter G of verifies :
How to build a barycenterTo draw it, decompose 2 of the 3 vectors according to the vectors you can already build. Long but simple :
So G is here :
Fundamental property of the barycenterYou've got to know that if G is the barycenter of the system , then for every point M in the plane, we have :
The drawing below with the figures at the beginning illustrates it well.
In practice and in problems you will usually have to place M at a particular place to prove things.
Coordinates of a barycenterIf A, B, and C are 3 points in an orthonormal frame, with , and if G is the barycenter of , then the folowing formulas give the coordinates of G :
You can use the formulas in case there are more than 3 points or in simple cases, with only 2 points. Finally you have to know that the barycenter of a system of points that all have same coefficients (the same weights) is called the isobarycenter.