Question 54
Let $I$ be the incentre of the acute triangle $ABC$. Rays $AI$ and $BI$ intersect the circumcircle $k$ of the triangle $ABC$ in points $D$ and $E$ respectively. Segments $DE$ and $CA$ intersect in point $F$, line through point $E$ parallel to the line $FI$ intersects circle $k$ also in point $G$, and lines $FI$ and $DG$ intersect in point $H$. Prove that the lines $CA$ and $BH$ touch the circumcircle of the triangle $DFH$ at points $F$ and $H$ respectively.