Question 5
Let $ABC$ be a triangle with $\angle ABC = 120^\circ$ and triangle bisectors $(AA_1)$, $(BB_1)$, $(CC_1)$, respectively. $B_1F \perp A_1C_1$, where $F \in (A_1C_1)$. Let $R$, $I$ and $S$ be the centers of circles which are inscribed in triangles $C_1B_1F$, $C_1B_1A_1$, $A_1B_1F$, and $B_1S \cap A_1C_1 = \{Q\}$. Show that $R$, $I$, $S$, $Q$ are on the same circle.