Proving the Centroid Theorem in Euclidean Geometry using Vector Techniques

[Christina The]

Homework Statement

Use vector techniques to prove the given theorem in Euclidean Geometry: A triangle and its medial triangle have the same centriod.

Homework Equations

The medial triangle of the triangle ABC is the triangle with vertices at the midpoints of the sides AB, AC, and BC of the triangle ABC. From an arbitrary point O that is not a vertex of the triangle ABC, the location of the centroid is given by (vector OA + vector OB + vector OC0/3

The Attempt at a Solution

x is midpoint of AB, y is midpoint of BC, z is midpoint of AC

from a point O (which I used the origin) the vectors of the sides of the triangle are as follows: AB = b-a , BC = c-b, and CA = a-c

the position vector for CX = CA + AX = (a-c) + 1/2 (b-a)
the position vector for AY = AB + BY = (b-a) + 1/2(c-b)
the position vector for BZ = CB + CZ = (c-b) + 1/2 (a-c)

vector equation fo the line CX = r = c + s(1/2(a+b)-c)

so, I can set up these vector equations...and I know that I am trying to show that the point at which the three vector equations intersect is the same as the point at which the vectors for the medials of the median triangle intersect. I don't know how to show how the vectors intersect.

 

[Eynstone]

The medial triangle has ( OA +OB)/2 &c. as vertices , which has the same average as OA,OB,OC. Hence the theorem.