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I would like some help about this:

Let there be 2 triangles $ABC$ and $DEF$. $A(2,0,1), B(3,2,-2), C(0,1,-1), D(-1,1,1), E(-3,2,0), F(0,1,1)$ and let $T_{1}$ and $T_{2}$ be centroids of triangle. Find $\overrightarrow{T_{1}T_{2}}$.

So centroid line(median) from point $A$ makes point $G$ which is middle point of line $BC$. It should have coordinates $$G = {1\over 2}(B+C) = \left({3\over 2},{3\over 2},-{3\over 2}\right)$$

So centroid line, in this case from point $A$ to $T_{1}$, is two times longer than $T_{1}$ to $G$. How can I find coordinate of $T_{1}$? Can i make vector $\overrightarrow{AG}$ and calculate $T_{1} = {1\over 3}(A+G)$ ?

Thanks for help!

Edit: Just found formula for finding coordinates of centroid, it should be $$T_{1} = {1\over 3}(x_{1}+x_{2}+x_{3},y_{1}+y_{2}+y_{3}, z_{1}+z_{2}+z_{3})$$ Could I solve this problem my or some other way?

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Consider this a start (i.e. a hint). To find the centroid of either triangle, use the definition.

Median. The median of a triangle is a line or line segment from a vertex to the midpoint of the opposite side.

Centroid. If three medians are constructed from the three vertices, they concur (meet) at a single point. That point is called the centroid. The centroid is a balance point for a triangle because all of the interior triangles that are formed have equal area.

So if 3 lines intersect at a point, then so 2 lines must intersect at the same point.

Step 1: Find the midpoint of segment AC. Let's call it $I=(1,.5,0)$

Step 2: Get an equation for the line between point B and opposite segment midpoint I. $(x,y,z)=B+\lambda(-2,-3/2,2)$ The direction vector, $(-2,-3/2,2)$ was obtained as $(I-B)$.

Step 3: Find the midpoint of segment CB. $J=(-4/3,4/3,2/3)$

Step 4: Get an equation for the line between point A and the opposite segment midpoint J. $(x,y,z)=A+\lambda(J-A)$ This one is $$ \left(\begin{array}{c} x\\ y\\ z \end{array}\right)=\left(\begin{array}{c} 2\\ 0\\ 1 \end{array}\right)+\lambda\left(\begin{array}{c} -\frac{1}{2}\\ \frac{3}{2}\\ -\frac{5}{2} \end{array}\right)$$

The intersection of these two lines is your centroid $T_1=(5/3, 1,-2/3)$. AND maybe I don't have numerical errors.

You can continue this way to get $T_2$ and then do whatever with them.

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