2021 IMO Shortlist G5

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

设 $A B C D$ 是一个循环四边形，其边长成对不同。令 $O$ 为 $A B C D$ 的外心。 $\angle A B C$ 和 $\angle A D C$ 的内角平分线分别在 $B_{1}$ 和 $D_{1}$ 处与 $A C$ 相交。令 $O_{B}$ 为穿过 $B$ 并在 $D_{1}$ 处与 $A C$ 相切的圆心。同样，设 $O_{D}$ 为穿过 $D$ 并在 $B_{1}$ 处与 $A C$ 相切的圆心。假设 $B D_{1} \| D B_{1}$。证明 $O$ 位于直线 $O_{B} O_{D}$ 上。