Problem

Source: Israel TST 2 2025 p3

Tags: geometry, cyclic quadrilateral, circumcircle, angle bisector



Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. The internal angle bisectors of \(\angle DAB\), \(\angle ABC\), \(\angle BCD\), \(\angle CDA\) create a convex quadrilateral $Q_1$. The external bisectors of the same angles create another convex quadrilateral $Q_2$. Prove $Q_1$, $Q_2$ are cyclic, and that $O$ is the midpoint of their circumcenters.