# Thread: Vector Proof for the Centroid of a triangle. Not a clue

1. ## Vector Proof for the Centroid of a triangle. Not a clue

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
I am to use vector techniques to prove that a triangle and its medial triangle have the same centroid, stating each step of the proof.
I don't know where to begin with Vector Proofs...

2. ## Re: Vector Proof for the Centroid of a triangle. Not a clue

Hello, sloanjg!

Here is a hint.

Code:
                A
o
*  *
*     *
*        *
D *           *  F
o  *  *  *  *  o
*  *           *  *
*     *        *     *
*        *     *        *
*            * *           *
B o  *  *  *  *  o  *  *  *  *  o C
E
We have triangle $ABC$ and medial triangle $DEF.$

Note that: . $FE \parallel AB$ and $FE = \tfrac{1}{2}AB$
. . That is: . $\overrightarrow{FE} \,=\,\tfrac{1}{2}\overrightarrow{AB}$
. . . .Also: . $\overrightarrow{DF} \,=\,\tfrac{1}{2}\overrightarrow{BC}$
. . . . And: . $\overrightarrow{DE} \,=\,\tfrac{1}{2}\overrightarrow{AC}$

Hope that helps . . .

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### prove that the centroid of a triangle.com

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