Let point I be the center of the inscribed circle of an acute-angled triangle ABC. Rays AI and BI intersect the circumcircle k of triangle ABC at points D and E respectively. The segments DE and CA intersect at point F, the line through point E parallel to the line FI intersects the circle k at point G, and the lines FI and DG intersect at point H. Prove that the lines CA and BH touch the circumcircle of the triangle DFH at the points F and H respectively. geometryincentercircumcircletangent