The theorem about concurrence of medians generalizes beautif

The theorem about concurrence of medians generalizes beautifully to three dimensions, where the figure corresponding to a triangle is a tetrahedron: a solid with four vertices joined by six lines that bound the tetrahedron\'s four triangular faces (Figure 4.7). 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. 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. 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. Deduce that the four lines from vertex to centroid of opposite face meet at the centroid of the tetrahedron.

Solution

4.3.2 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. Now, 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=1/2(v+w)u.Then u+a=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+2/3a also is a point on the median that passes through u. But u+2/3 * a = u + 2/3 * 1/2 *((v+w)-u))=1/3 *(u+v+w) 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.