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Problem
Problem
Consider a
Δ
A
B
C
\Delta ABC
Δ
A
B
C
. A directly similar
Δ
A
1
B
1
C
1
\Delta A_1 B_1 C_1
Δ
A
1
B
1
C
1
is inscribed in the
Δ
A
B
C
\Delta ABC
Δ
A
B
C
such that
A
1
,
B
1
A_1, B_1
A
1
,
B
1
and
C
1
C_1
C
1
are the interior points of the sides
A
C
,
A
B
AC, AB
A
C
,
A
B
and
B
C
BC
B
C
, respectively. Prove that
Area
(
Δ
A
1
B
1
C
1
)
Area
(
Δ
A
B
C
)
≥
1
csc
2
A
+
csc
2
B
+
csc
2
C
\frac{\text{Area}(\Delta A_1 B_1 C_1)}{\text{Area}(\Delta ABC)} \ge \frac{1}{\csc^2 A + \csc^2 B + \csc^2 C}
Area
(
Δ
A
B
C
)
Area
(
Δ
A
1
B
1
C
1
)
≥
c
s
c
2
A
+
c
s
c
2
B
+
c
s
c
2
C
1
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