Mathematics

The key insight is that the centroid is the average position of the three vertices—so the displacement of the centroid depends only on the average displacement of the vertices. Express both centroids using position vectors ($G = \frac{\vec{A} + \vec{B} + \vec{C}}{3}$ and similarly for $G'$), then compute $\vec{GG'}$ and relate it to the individual displacements $\vec{AA'}$, $\vec{BB'}$, and $\vec{CC'}$. The algebra will reveal that these three displacement vectors sum to exactly three times the centroid displacement.

The key insight is that the centroid is the average position of the three vertices—so the displacement of the centroid depends only on the average displacement of the vertices. Express both centroids using position vectors ($G = \frac{\vec{A} + \vec{B} + \vec{C}}{3}$ and similarly for $G'$), then compute $\vec{GG'}$ and relate it to the individual displacements $\vec{AA'}$, $\vec{BB'}$, and $\vec{CC'}$. The algebra will reveal that these three displacement vectors sum to exactly three times the centroid displacement.