Suppose that the tetrahedron has vertices t, u, v, and w. Show that the centroid of the face opposite to t is 1/3 (u+v+w)

Question

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Suppose that the tetrahedron has vertices t, u, v, and 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.

Answer 1

The face opposite to $t$ is the triangle with vertices $u$, $v$, and $w$. The other three faces also are triangles, one for each of the other sets of three vertices you can make from $t$, $u$, $v$, and $w$.

I'll outline some steps of a proof. You may want to stop reading at some point and see how many of the remaining steps you can intuit.

The centroid of a triangle is at the intersection of the three medians. The median through one vertex of the triangle also passes through the midpoint of the opposite side. Given a vertex, we can express the midpoint of the opposite side in terms of its two vertices, and we can then express a point 2/3 of the way from the given vertex to the midpoint of the opposite side in terms of the three vertices. That point is the intersection of the three medians, but we can use this construction to prove that fact.

For the triangle with vertices at $u$, $v$, and $w$, let's do the median through $u$ first. Let

$$a = \frac{1}{2}(v + w ) - u.$$

Then $u + a = \frac{1}{2}(v + w)$, which is the midpoint of the side with vertices $v$ and $w$, so $u$ and $u + a$ are both on the triangle's median that passes through $u$, and therefore $u + \frac{2}{3} a$ also is a point on the median that passes through $u$. But

\begin{eqnarray} u + \frac{2}{3} a & = & u + \frac{2}{3}\left(\frac{1}{2}(v + w) - u\right) \\ &= & \frac{1}{3}(u + v + w) \end{eqnarray}

Perform a similar procedure for the median through $v$ and the median through $w$. The result will be the same point. You then will have shown that this same point is at the intersection of the medians (and incidentally you will have proved that the three medians do in fact intersect at a single point) and therefore it is the centroid of the triangle.

Answer 2

The centroids of the other three faces are $1/3 (t+u+v)$, $1/3 (t+u+w)$, and $1/3 (t+v+w)$. The proof, however, is something that is unclear to me.