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National and Regional Contests
India Contests
Regional Mathematical Olympiad
2010 India Regional Mathematical Olympiad
1
1
Part of
2010 India Regional Mathematical Olympiad
Problems
(1)
Hexagon
Source:
12/5/2010
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon in which diagonals
A
D
,
B
E
,
C
F
AD, BE, CF
A
D
,
BE
,
CF
are concurrent at
O
O
O
. Suppose
[
O
A
F
]
[OAF]
[
O
A
F
]
is geometric mean of
[
O
A
B
]
[OAB]
[
O
A
B
]
and
[
O
E
F
]
[OEF]
[
OEF
]
and
[
O
B
C
]
[OBC]
[
OBC
]
is geometric mean of
[
O
A
B
]
[OAB]
[
O
A
B
]
and
[
O
C
D
]
[OCD]
[
OC
D
]
. Prove that
[
O
E
D
]
[OED]
[
OE
D
]
is the geometric mean of
[
O
C
D
]
[OCD]
[
OC
D
]
and
[
O
E
F
]
[OEF]
[
OEF
]
. (Here
[
X
Y
Z
]
[XYZ]
[
X
Y
Z
]
denotes are of
△
X
Y
Z
\triangle XYZ
△
X
Y
Z
)
trigonometry
geometry