Assume that u, v, and w are vectors in R³ that form the sides of a triangle (see figure). Use the following steps to prove that the medians intersect at a point that divides each median in a 2:1 ratio. The proof does not use a coordinate system.

a. Show that u + v + w = 0.

b. Let M₁ be the median vector from the midpoint of u to the opposite vertex. Define M₂ and M₃ similarly. Using the geometry of vector addition show that M₁ = u/2 + v. Find analogous expressions for M₂ and M₃.

c. Let a, b, and c be the vectors from O to the points one-third of the way along M₁, M₂, and M₃, respectively. Show that a = b = c = (u - w)/3.

d. Conclude that the medians intersect at a point that divides each median in a 2:1 ratio.

Transcribed Image Text:

M, M2 M, 2.