Distance between Centroids of the Faces of a Regular Tetrahedron

Construct a cube with side length $\frac {\sqrt 2}{2}$

choose one vertex. There are 3 vertexes that are diagonally across each face.

Join these 4 vertexes.

You have constructed a regular tetrahedron with side length equal to 1.

See figure.

I have marked the centroid of two faces with the little dots.

Now lets move to the top view.... The centroid is $\frac 23$ the distance from the vertex to the opposite edge. Which means that the distance between the centroid of two faces is $\frac 13$ the lenght of the diagonal of the face. And we have already discussed that that length is 1.

$\frac 13$