  We consider the variant of the Banach-Mazur distance $\delta_{BM}^{\rm cen}
(C, D)$ of two convex bodies $C, D$ of $E^d$ with the additional requirement
that the centroids of them coincide. We prove that $\delta_{BM}^{\rm cen} (C,
D) \leq \frac{69}{17}$ for every $C, D$ of $E^2$.
