Question 98
Let $\Gamma$ be the circumcircle of a triangle $ABC$, and $\ell$ be the line tangent to $\Gamma$ at point $A$. Let $D, E$ be interior points of the sides $AB, AC$, respectively, satisfying the condition $BD : DA = AE : EC$. Let $F, G$ be the two points of intersection of line $DE$ and circle $\Gamma$, $H$ be the point of intersection of the line $\ell$ and the line parallel to $AC$ and going through point $D$, and $I$ be the point of intersection of the line $\ell$ and the line parallel to $AB$ and going through point $E$. Prove that four points $F, G, H, I$ lie on the circumference of a circle which is tangent to line $BC$. Here, for a line segment $XY$ we denote its length also by $XY$.
