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Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2008 Korean National Olympiad
6
6
Part of
2008 Korean National Olympiad
Problems
(1)
2008 KMO P6
Source:
8/9/2015
Let
A
B
C
D
ABCD
A
BC
D
be inscribed in a circle
ω
\omega
ω
. Let the line parallel to the tangent to
ω
\omega
ω
at
A
A
A
and passing
D
D
D
meet
ω
\omega
ω
at
E
E
E
.
F
F
F
is a point on
ω
\omega
ω
such that lies on the different side of
E
E
E
wrt
C
D
CD
C
D
. If
A
E
⋅
A
D
⋅
C
F
=
B
E
⋅
B
C
⋅
D
F
AE \cdot AD \cdot CF = BE \cdot BC \cdot DF
A
E
⋅
A
D
⋅
CF
=
BE
⋅
BC
⋅
D
F
and
∠
C
F
D
=
2
∠
A
F
B
\angle CFD = 2\angle AFB
∠
CF
D
=
2∠
A
FB
, Show that the tangent to
ω
\omega
ω
at
A
,
B
A, B
A
,
B
and line
E
F
EF
EF
concur at one point. (
A
A
A
and
E
E
E
lies on the same side of
C
D
CD
C
D
)
geometry
concurrence