![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
is a triangle, the bisector of angle
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
meets the circumcircle of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
in
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
, points
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
and
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
are defined similarly. Let
![AA_1](/media/m/e/3/4/e344b2dabbe38d5029e9c27e6a129e81.png)
meet the lines that bisect the two external angles at
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
in
![A_0](/media/m/2/f/f/2ff029e43a310c5b6d8137a3edb7609c.png)
. Define
![B_0](/media/m/5/a/7/5a7b148f9ae7eef70595a0deebfddd3a.png)
and
![C_0](/media/m/e/9/e/e9eb7207b3e27429b1d887f6793224be.png)
similarly. Prove that the area of triangle
![A_0B_0C_0 = 2 \cdot](/media/m/6/c/4/6c436f6e5ef3f169bdf8cd674d3e6f59.png)
area of hexagon
![AC_1BA_1CB_1 \geq 4 \cdot](/media/m/a/0/d/a0da971cc47775a119912015deadb048.png)
area of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
.
%V0
$ABC$ is 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 $A_0$. Define $B_0$ and $C_0$ similarly. Prove that the area of triangle $A_0B_0C_0 = 2 \cdot$ area of hexagon $AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ABC$.