Problem

Source: ELMO 2014 Shortlist G2, by Yang Liu

Tags: geometry, circumcircle, Asymptote, cyclic quadrilateral, power of a point, radical axis, projective geometry



$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$. Let $AB \cap CD = E$, $AD \cap BC = F$. Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$, respectively. Let $\omega \cap \omega_1 = G$, $\omega \cap \omega_2 = H$. Show that $AC, BD, GH$ are concurrent. Proposed by Yang Liu