Question 175
From a point $A$ lying outside the circle $(O)$, draw two tangent lines $AB$, $AC$ of $(O)$ with $B$, $C$ are tangent points. A line passes through $A$, lies inside the angle $OAC$, cuts $(O)$ at $R$, $S$ ($R$ is between $A$ and $S$). The segments $BR$, $BS$ cut the ray $AO$ respectively at $D$, $E$. Denote $H$ as orthocenter and $I$ as circumcenter of triangle $BDE$. Let $BT$ be the diameter of circumcircle of $(I)$. Prove that $\triangle DHT \sim \triangle RBS$ and calculate the ratio that $OI$ divides $BC$.