When projecting the vertices $A$, $B$ and $C$ of a traingle onto a plane, getting the points $A'$, $B'$ and $C'$, is the projection of $G$ (barycenter of the triangle $ABC$) the same as the barycenter of $A'B'C'$?
I think the way is showing that the projection of the midpoints of the sides of the triangle are the midpoints of the projected sides.
The answer is yes, and the reason is both deep and straightforward.
A projection onto a plane is an affine map. This means by definition that it respects all affine combinations, and barycentres are a special kind of affine combinations (where all weights are equal).
An easy way to see why such a projection is affine is that if you choose the origin of $\Bbb R^n$ carefully (namely, on the plane onto which you are projecting), then the projection is a linear map. Linear maps are trivially affine.