Question 4
Let $ABC$ be a triangle with circumcircle $\omega$. Let $D$, $E$ and $F$ be points on the sides $BC$, $CA$ and $AB$ such that the circumcircle of the triangle $DEF$ touches $\omega$ at $A$. Let $G$ and $H$ be the intersection points of the circumcircles of the triangles $BDE$ and $CDF$ with $\omega$ (different from $B$, $C$), respectively. Prove that the lines $GE$ and $HF$ intersect on $AD$.
