IMO Shortlist 2021 G5
IMO Shortlist 2021 G5 is a plane geometry problem from the IMO Shortlist 2021, Geometry G5: the circumcenter on the line through two circle centres.
Problem
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 $BD_1 \parallel DB_1$. Prove that $O$ lies on the line $O_BO_D$.
Source IMO Shortlist 2021, Geometry G5
The diagram
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