Question 2
Let $ABC$ be a triangle. The bisector of angle $A$ meets the circumcircle of triangle $ABC$ in $A_1$. Points $B_1$ and $C_1$ are defined similarly. Let $AA_1$ meet the lines that bisect the two external angles at $B$ and $C$ in point $A^0$. Define $B^0$ and $C^0$ similarly. If $S_{X_1X_2\ldots X_n}$ denotes the area of the polygon $X_1X_2\ldots X_n$, prove that
$$S_{A^0B^0C^0}=2S_{AC_1BA_1CB_1}\ge4S_{ABC}.$$
$$S_{A^0B^0C^0}=2S_{AC_1BA_1CB_1}\ge4S_{ABC}.$$