The centers of gravity of the associahedron and of the permutahedron are the same

In this article, we show that Loday’s realization of the associahedron has the the same center of gravity than the permutahedron. This proves an observation made by F. Chapoton. We also prove that this result holds for the associahedron and the cyclohedron as realized by the first author and C. Lange.