JBMO-SL 2008 G9

Let $O$ be a point inside the parallelogram $A B C D$ such that

$$\angle A O B+\angle C O D=\angle B O C+\angle C O D$$

Prove that there exists a circle $k$ tangent to the circumscribed circles of the triangles $\triangle A O B, \triangle B O C, \triangle C O D$ and $\triangle D O A$.