Question 28
We are given an arbitrary acute-angled triangle $ABC$ and its altitudes $AD$ and $BE$ where $D$ and $E$ denote their feet on sides $BC$ and $AC$, respectively. Let furthermore $F$ and $G$ be two points on segments $AD$ and $BE$, respectively, such that
$$
\frac{AF}{FD} = \frac{BG}{GE}.
$$
The line through $C$ and $F$ intersects $BE$ in point $H$ and the line through $C$ and $G$ intersects $AD$ in point $I$. Prove that the four points $F, G, H$ and $I$ are concyclic.
$$
\frac{AF}{FD} = \frac{BG}{GE}.
$$
The line through $C$ and $F$ intersects $BE$ in point $H$ and the line through $C$ and $G$ intersects $AD$ in point $I$. Prove that the four points $F, G, H$ and $I$ are concyclic.