I've annotated the image below. $K$ is the triangle we start with, K1,K2,K3 are the three equilateral triangles constructed around it, and the solid green triangle is the triangle having the three centroids of K1,K2,K3 as vertices
Now the idea is that the green line connecting the centroid of K1 and the centroid of K2 has the same length as the dotten green line connecting the centroids of K2 and K1'. This is the case because the two lines are constructed in exactly the same way, only that $K1'$ takes the place of $K1$ and K3' takes the place of K3.
The same reasoning works for all the other dotted green lines, which shows that these lines form an equilateral hexagon.
Call the remaining vertices of the hexagon $K2', K1'',$ and $K2''$, so that the hexagon's vertices read $K1K2K1'K2'K1''K2''$ in clockwise order.
Now consider the angles of this hexagon. The angles at $K1$, $K1'$, and $K1''$ are equal by construction. The angles at $K2$, $K2'$, and $K2''$ are also equal by construction. Then notice that this hexagon, when translated (but not rotated!), tiles the plane in the way shown in the diagram. Therefore, looking at vertex $K1'$ where three hexagons meet, we have that angles $K1$, $K1'$, and $K1''$ must add up to $360^\circ$. Therefore each is equal to $120^\circ$. It follows that $K2$, $K2'$, and $K2''$ are also equal to $120^\circ$, so the hexagon is regular.
Since the hexagon is regular, it lies on a circle. Since $K1K3 = K1'K3 = K1''K3$ (by construction), the center of the circle is $K3$.
Finally, for a regular hexagon, the sides have the same length as the radius, so we conclude
K1K3 = K2K3 = K1K2
which is the desired result.
Hardly a full proof without words when it requires so much justification, but perhaps the "proof without words" was more of a visualization. Or perhaps there is a simpler way to reason this through using the fact that the hexagon tiles the plane directly.