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Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2019 Korea National Olympiad
2
2
Part of
2019 Korea National Olympiad
Problems
(1)
Circle centered on A-excenter passing A
Source: Korea National Olympiad 2019 P2
11/16/2019
Triangle
A
B
C
ABC
A
BC
is an scalene triangle. Let
I
I
I
the incenter,
Ω
\Omega
Ω
the circumcircle,
E
E
E
the
A
A
A
-excenter of triangle
A
B
C
ABC
A
BC
. Let
Γ
\Gamma
Γ
the circle centered at
E
E
E
and passes
A
A
A
.
Γ
\Gamma
Γ
and
Ω
\Omega
Ω
intersect at point
D
(
≠
A
)
D(\neq A)
D
(
=
A
)
, and the perpendicular line of
B
C
BC
BC
which passes
A
A
A
meets
Γ
\Gamma
Γ
at point
K
(
≠
A
)
K(\neq A)
K
(
=
A
)
.
L
L
L
is the perpendicular foot from
I
I
I
to
A
C
AC
A
C
. Now if
A
E
AE
A
E
and
D
K
DK
DK
intersects at
F
F
F
, prove that
B
E
⋅
C
I
=
2
⋅
C
F
⋅
C
L
BE\cdot CI=2\cdot CF\cdot CL
BE
⋅
C
I
=
2
⋅
CF
⋅
C
L
.
geometry
circumcircle