### (IMO 1989-2) Area of $A_0B_0C_0$

Posted:

**Wed Feb 01, 2012 12:01 am**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 $S_{X_1X_2...X_n}$ denotes the area of the polygon

$X_1 X_2 . . . X_n$ , prove that,

$S_{A_0B_0C_0}=2S_{AC_1BA_1CB_1}≥ 4S_{ABC}$.

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 $S_{X_1X_2...X_n}$ denotes the area of the polygon

$X_1 X_2 . . . X_n$ , prove that,

$S_{A_0B_0C_0}=2S_{AC_1BA_1CB_1}≥ 4S_{ABC}$.