The lines joining the vertices of a tetrahedron to the centroids of opposite faces are concurrent.

Explanation

Let `{ABCD}` be a tetrahedron such that the coordinates of its vertices are

`A(x y_{1}, z_{1}), B(x_{2}, y_{2}, z_{2}), C(x_{3}, y_{3}, z_{3})` and `D(x_{4}, y_{4}, z_{4})`. The coordinates of the centroids of faces `{ABC}, {DAB}, {DBC}` and DCA are respectively

`{G}_{1}(frac{{x}_{1}+{x}_{2}+{x}_{3}}{3}, frac{{y}_{1}+{y}_{2}+{y}_{3}}{3}, frac{{z}_{1}+{z}_{2}+{z}_{3}}{3})`

`{G}_{2}(frac{{x}_{1}+{x}_{2}+{x}_{4}}{3}, frac{{y}_{1}+{y}_{2}+{y}_{4}}{3}, frac{{z}_{1}+{z}_{2}+{z}_{4}}{3})`

`{G}_{3}(frac{{x}_{2}+{x}_{3}+{x}_{4}}{3}, frac{{y}_{2}+{y}_{3}+{y}_{4}}{3}, frac{{z}_{2}+{z}_{3}+{z}_{4}}{3})`

and, `{G}_{4}(frac{{x}_{4}+{x}_{3}+{x}_{1}}{3}, frac{{y}_{4}+{y}_{3}+{y}_{1}}{3}, frac{{z}_{4}+{z}_{3}+{z}_{1}}{3})`

Now, coordinates of point `{G}` dividing `{DG}_{1}` in the ratio `3: 1` are
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