Problem

Source: Indonesian Stage 1 TST for IMO 2022, Test 3 (Geometry)

Tags: geometry, diameter, circumcircle, Concyclic



Let $AB$ be the diameter of circle $\Gamma$ centred at $O$. Point $C$ lies on ray $\overrightarrow{AB}$. The line through $C$ cuts circle $\Gamma$ at $D$ and $E$, with point $D$ being closer to $C$ than $E$ is. $OF$ is the diameter of the circumcircle of triangle $BOD$. Next, construct $CF$, cutting the circumcircle of triangle $BOD$ at $G$. Prove that $O,A,E,G$ are concyclic. (Possibly proposed by Pak Wono)