Question 30
Let $\triangle ABC$ be a triangle with circumcircle $\Gamma$, and let $I$ be the center of the incircle of $\triangle ABC$. The lines $AI$, $BI$ and $CI$ intersect $\Gamma$ in $D \neq A$, $E \neq B$ and $F \neq C$. The tangent lines to $\Gamma$ in $F$, $D$ and $E$ intersect the lines $AI$, $BI$ and $CI$ in $R$, $S$ and $T$, respectively. Prove that
$$
|AR| \cdot |BS| \cdot |CT| = |ID| \cdot |IE| \cdot |IF|
$$
$$
|AR| \cdot |BS| \cdot |CT| = |ID| \cdot |IE| \cdot |IF|
$$