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Balkan MO
1992 Balkan MO
3
3
Part of
1992 Balkan MO
Problems
(1)
Three points on the sides of triangle ABC imply inequality
Source: Balkan MO 1992, Problem 3
4/25/2006
Let
D
D
D
,
E
E
E
,
F
F
F
be points on the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
respectively of a triangle
A
B
C
ABC
A
BC
(distinct from the vertices). If the quadrilateral
A
F
D
E
AFDE
A
F
D
E
is cyclic, prove that
4
A
[
D
E
F
]
A
[
A
B
C
]
≤
(
E
F
A
D
)
2
.
\frac{ 4 \mathcal A[DEF] }{\mathcal A[ABC] } \leq \left( \frac{EF}{AD} \right)^2 .
A
[
A
BC
]
4
A
[
D
EF
]
≤
(
A
D
EF
)
2
.
Greece
inequalities
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circumcircle
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