Question 73
Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcentre of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_{1}$ and $D_{1}$, respectively. Let $O_{B}$ be the centre of the circle which passes through $B$ and is tangent to $AC$ at $D_{1}$. Similarly, let $O_{D}$ be the centre of the circle which passes through $D$ and is tangent to $AC$ at $B_{1}$.
Assume that $B D_{1} \parallel D B_{1}$. Prove that $O$ lies on the line $O_{B} O_{D}$.
Assume that $B D_{1} \parallel D B_{1}$. Prove that $O$ lies on the line $O_{B} O_{D}$.