Consider those real triangles whose vertices have coordinates from the numbers $-1,0,1$ in a plane Cartesian coordinate system. How far can the centroid of these triangles be from the origin?
Let the coordinates of the vertices of triangle $ABC$ be $A\left(a_{1} ; a_{2}\right), B\left(b_{1} ; b_{2}\right), C\left(c_{1} ; c_{2}\right)$.

We know that the coordinates $\left(s_{1} ; s_{2}\right)$ of the centroid of the triangle can be expressed in terms of the coordinates of the vertices:

$$
s_{1}=\frac{a_{1}+b_{1}+c_{1}}{3}, \quad s_{2}=\frac{a_{2}+b_{2}+c_{2}}{3}
$$

The possible values of $a_{1}+b_{1}+c_{1}$ and $a_{2}+b_{2}+c_{2}$ are $-3,-2,-1,0,1,2,3$, but in the case of $\pm 3$, a real triangle does not form, so we only need to consider the other cases.

The distance of the centroid from the origin is $d=\sqrt{s_{1}^{2}+s_{2}^{2}}$, where the values of $3 s_{1}$ and $3 s_{2}$ are from the set $\{0, \pm 1, \pm 2\}$.

These yield the following coordinate pairs:

$S: \quad(0 ; 0), \quad\left(0 ; \pm \frac{1}{3}\right), \quad\left(0 ; \pm \frac{2}{3}\right), \quad\left( \pm \frac{1}{3} ; \pm \frac{1}{3}\right), \quad\left( \pm \frac{1}{3} ; \pm \frac{2}{3}\right), \quad\left( \pm \frac{2}{3} ; \pm \frac{2}{3}\right)$,

$S: \quad(0 ; 0), \quad\left(0 ; \pm \frac{1}{3}\right), \quad\left(0 ; \pm \frac{2}{3}\right), \quad\left( \pm \frac{1}{3} ; \pm \frac{1}{3}\right), \quad\left( \pm \frac{1}{3} ; \pm \frac{2}{3}\right), \quad\left( \pm \frac{2}{3} ; \pm \frac{2}{3}\right)$,

and their distances from the origin are respectively: $0, \quad \frac{1}{3}, \quad \frac{2}{3}, \quad \frac{\sqrt{2}}{3}, \quad \frac{\sqrt{5}}{3}, \quad \frac{2 \sqrt{2}}{3}$.
![](https://cdn.mathpix.com/cropped/2024_05_02_ceec02b784e2a7a25cecg-1.jpg?height=1444&width=1482&top_left_y=830&top_left_x=300)

It is easy to see that each value corresponds to a triangle (not just one). The diagrams show the triangles where neither coordinate of the centroid is negative.

If we reflect these triangles across one of the coordinate axes, i.e., change the sign of one of the coordinates of the vertices, the distance of the centroid from the origin does not change. Therefore, the only possible values for the distances are indeed the six listed values.