Question 12
Let $ABCD$ be a quadrilateral circumscribed about a circle with center $O$. Let $O_{1}, O_{2}, O_{3}$, and $O_{4}$ denote the circumcenters of $\triangle AOB$, $\triangle BOC$, $\triangle COD$, and $\triangle DOA$. If $\angle A = 120^{\circ}$, $\angle B = 80^{\circ}$, and $\angle C = 45^{\circ}$, what is the acute angle formed by the two lines passing through $O_{1}O_{3}$ and $O_{2}O_{4}$?
