The medial triangle

Vector techniques can be used to prove and derive many geometric relationships. Use the following theorem from Euclidean geometry as you complete this assessment:


Theorem: A triangle and its medial triangle have the same centroid.

Scenario:

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.