I saw on Twitter this cool mechanical linkage for which the red dot corresponds to the centroid of the triangle defined by the blue dots:
Can you prove why this linkage works?
The proof is in two parts, corresponding to the two linkages which are joined to each other at a single point. For each part, I'll try to both explain in words and illustrate on the picture you've provided.
The first linkage, on the left in your picture, provides the midpoint between the two blue points on this linkage, because the two short legs are of equal length. Joining that midpoint to the third blue point gives one median of the triangle. (The centroid of a triangle is the intersection of its medians.)
The second linkage, on the right in your picture, provides the centroid because the two short legs are of different lengths, one twice as long as the other, so their meeting point is twice as far from the third blue point as from the previously discovered midpoint. (The centroid of a triangle is two-thirds of the way along each median.)
The mechanism that resembles a smaller V combined with a larger V is known as a pantograph. In this case, it has the joint of the smaller V maintain a fixed ratio of distance along the line segment between the two endpoints. As the accepted answer notes, the two mechanisms show two thirds the way along a triangle's median as the centroid.
Proof sketch of Centroid Theorem (this is a good geometry puzzle)
By definition of centroid, it must lie along all three medians, so it is at the intersection of the medians. This can be proved to exist by constructing two medians and then showing the third goes through the intersection. By constructing all three medians and using the areas of the six triangles and the definition of median (splitting a side in half), the centroid can be shown to be two thirds the way along a median.