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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.

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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.

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A vertex figure is made by connecting the midpoints of all the edges which touch a given vertex.

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(3) Common notation for describing regular or semiregular tessellations uses vertex arrangements, which list the number of sides of each polygon going around the vertex. Give the vertex arrangements for the regular and semiregular tessellations.

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