BALKAN 2009 Problem 3

A $9 \times 12$ rectangle is divided into unit squares. The centers of all the unit squares, except the four corner squares and the eight squares adjacent (by side) to them, are colored red. Is it possible to numerate the red centers by $C_{1}, C_{2}, \ldots, C_{96}$ so that the following two conditions are fulfilled: $1^{\circ}$ All segments $C_{1} C_{2}, C_{2} C_{3}, \ldots C_{95} C_{96}, C_{96} C_{1}$ have the length $\sqrt{13}$; $2^{\circ}$ The poligonal line $C_{1} C_{2} \ldots C_{96} C_{1}$ is centrally symmetric? (Bulgaria)