Position of the centroid of a planar convex body

It is well known that any planar convex body A permits to inscribe an affine-regular hexagon \(H_A\) $H_A$ . We prove that the centroid of A belongs to the homothetic image of \(H_A\) $H_A$ with ratio \(\frac{4}{21}\) $\frac{4}{21}$ and the center in the center of \(H_A\) $H_A$ . This ratio cannot be decreased.