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    Centroids of triangles

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    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.

    © BrainMass Inc. brainmass.com May 20, 2020, 11:40 pm ad1c9bdddf
    https://brainmass.com/math/geometry-and-topology/centroids-of-triangles-584627

    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.

    $2.19

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