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Circumcenters are collinear under parallel condition

Source: ISL 2021 G5

July 12, 2022
geometrycircumcircleprojective geometry

Problem Statement

Let ABCDABCD be a cyclic quadrilateral whose sides have pairwise different lengths. Let OO be the circumcenter of ABCDABCD. The internal angle bisectors of ∠ABC\angle ABC and ∠ADC\angle ADC meet ACAC at B1B_1 and D1D_1, respectively. Let OBO_B be the center of the circle which passes through BB and is tangent to AC‾\overline{AC} at D1D_1. Similarly, let ODO_D be the center of the circle which passes through DD and is tangent to AC‾\overline{AC} at B1B_1.
Assume that BD1‾∥DB1‾\overline{BD_1} \parallel \overline{DB_1}. Prove that OO lies on the line OBOD‾\overline{O_BO_D}.
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