BALKAN-SL 2009 C1

A $9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $\sqrt {13}$, ii) the closed broken line $C_1C_2\ldots C_{96}C_1$ has a centre of symmetry?

Bulgaria