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Consider a . A directly similar is inscribed in such that , , and are interior points…
MATHEMATICS · Subjective · CBSE · Mathematics · Triangles · Triangles
Problem
Consider a
△
A
B
C
\triangle ABC
△
A
B
C
. A directly similar
△
A
1
B
1
C
1
\triangle A_1B_1C_1
△
A
1
B
1
C
1
is inscribed in
△
A
B
C
\triangle ABC
△
A
B
C
such that
A
1
A_1
A
1
,
B
1
B_1
B
1
, and
C
1
C_1
C
1
are interior points of
A
C
AC
A
C
,
A
B
AB
A
B
, and
B
C
BC
B
C
, respectively. Prove that
area
(
△
A
1
B
1
C
1
)
area
(
△
A
B
C
)
≥
1
csc
2
A
+
csc
2
B
+
csc
2
C
\frac{\operatorname{area}(\triangle A_1B_1C_1)}{\operatorname{area}(\triangle ABC)}\geq\frac{1}{\csc^2A+\csc^2B+\csc^2C}
area
(
△
A
B
C
)
area
(
△
A
1
B
1
C
1
)
≥
c
s
c
2
A
+
c
s
c
2
B
+
c
s
c
2
C
1
.
Related theory
Similarity of Triangles
Similarity
Areas of Similar Triangles
Triangles
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