# Centroids of triangles

The medial triangle of a triangle ABC is the triangle whose vertices are located at the midpoints of the sides AB, AC, and BC of triangle ABC. From an arbitrary point O that is not a vertex of triangle ABC, you may take it as a given fact that the location of the centroid of triangle ABC is the vector (vector OA + vector OB + vector OC)/3.

Task:

A. Use vector techniques to prove that a triangle and its medial triangle have the same centroid, stating each step of the proof.

1. Provide written justification for each step of your proof.

B. Provide a convincing argument short of a proof (suggested length of 3-4 sentences) that the theorem is true.

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#### Solution Preview

A. Let us assume mid points of AB, BC and CA are L, M and N respectively.

Let us assume centroid of ABC is G, and of LMN is G'.

Because,

vector(OG) = (vector(OA) + vector(OB) + vector(OC))/3

L is mid point of AB

vector(OL) = vector(OA) + vector(AL)

As AL = LB = AB/2

Hence,

vector(OL) = vector(OA) + vector(AB/2)

=> vector(OL) = ...

#### Solution Summary

This solution provides proof that the centroid of a triangle and the centroid of medial triangle coincide.