supposed that the tetrahedron has verticies t, u, v, w. Show that the centroid o

Question

supposed that the tetrahedron has verticies t, u, v, w. Show that the centroid of the face opposite to t is 1/3(u+v+w) and write down the centroids of the other three faces

now consider each line joining a vertex to the centroid of the opposite face. in particular, show that the point 3/4 of the way from t to the centroid of the opposite face is 1/4(t+u+v+w)--the centroid of the tetrahedron

briefly explain why the point 1/4(t+u+v+w) lies on the other three lines from a vertex to the centroid of the opposite face

I AM SUPPOSED TO USE GEOMETERS SKETCHPAD FOR THIS PROBLEM AND THIS IS WHAT IVE DONE SO FAR...AM I ON THE RIGHT TRACK?

Explanation / Answer

You can easily answer this question with a similar argument to that of concurrence of medians of a triangle in R2. The centroid of each face is 1/3 times its vertices, so if we take the line from t(assuming it is the top most vertice of the tetrahedron, and think about the point 3/4 of the way down to the centroid of the face, we get:

vertice ,t, to the point 3/4 to the centroid(1/3(u+v+w)), and finally subract off the 3/4t so we dont double count. Thus we get:

t + 3/4(1/3(u+v+w) - t)

= t + (1/4u + 1/4v + 1/4w) - 3/4t

= 1/4u + 1/4v + 1/4w + 1/4t

=1/4(t+u+v+w)

and finally, this point, is the centroid of the tetrahedron itself, so expanding concurrence of medians to R3, if all lines from vertices to midpoints of opposite FACES(in R3) intersect at the same point, mainly the centroid, then this pointy lies in all lines from vertices to centroids of opposite faces

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