MathNet 2023-part-05 — 191 problems (EN–VI) Question 155

Let $ABCD$ be a cyclic quadrilateral. Take $F \in AB$, $E \in CD$ such that $(FCD)$ is tangent to $AB$ and $(EAB)$ is tangent to $CD$. Denote $G = AE \cap DF$, $H = BE \cap CF$. Prove that $EF$ is the perpendicular bisector of $GH$.