Does the sum of three vectors originating from the centroid of a triangle and pointing to the angles always sum to zero?

Question

Consider an equilateral triangle as the one given in the figure below:

Assume $$\vec{E_1},\vec{E_2},\vec{E_3}$$ are vectors (for instance complex numbers in the complex plane) originating from the centroid (N) and pointing each one to their respective angle (each vector is parallel to a median).

For an equilateral triangle, the property (property 1):

$$\vec{E_1}+\vec{E_2}+\vec{E_3}=0 $$ holds true.

Now consider a generic triangle where the three vectors still originate at the centroid (N). My question is: is the above property (property 1) still true for a generic triangle?

Additional information: this question is related to an electrotechnical problem and to a question I originally asked on Electronics SE. As far as my understanding goes, this geometrical property is the key to answering the electrotechnical question, therefore I thought it could be appropriate to ask it here. If you wish to know more about the original question please let me know.

Answer 1

Here's a proof: By a preexisting formula, if the coordinates of the vertices of a triangle are $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, then the centroid is the point $$(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})$$ If this is so, then the horizontal parts of each of the vectors are $\frac{-2x_1+x_2+x_3}{3}$, $\frac{x_1-2x_2+x_3}{3}$, and $\frac{x_1+x_2-2x_3}{3}$, and the vertical parts are $\frac{-2y_1+y_2+y_3}{3}$, $\frac{y_1-2y_2+y_3}{3}$, and $\frac{y_1+y_2-2y_3}{3}$. To sum the vectors, sum the horizontal and vertical parts. The horizontal part will be $$\frac{-2x_1+x_2+x_3}{3}+\frac{x_1-2x_2+x_3}{3}+\frac{x_1+x_2-2x_3}{3} = 0$$ Which is $0$, and the vertical part will be $$\frac{-2y_1+y_2+y_3}{3}+\frac{y_1-2y_2+y_3}{3}+\frac{y_1+y_2-2y_3}{3} = 0$$ Which is also $0$. All $3$ vectors cancel out. QED.

Answer 2

Notice that the property you wish to prove is an affine property. So, yes, if it holds for a single triangle it holds for all triangles.

In more detail, an affine transformation is determined by where any three noncollinear points get mapped to, and since the vertices of a triangle are such, then any two triangles are affine equivalent.

The centroid (or barycenter) of any finite set of points is a particular weighted sum of the points with all weights equal, and the sum of vectors from the centroid to the points is zero while weighted sums are an affine property.

Some good information is at https://en.wikipedia.org/wiki/Affine_transformation.