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Korea Junior Mathematics Olympiad
2020 Korea Junior Math Olympiad
2
2
Part of
2020 Korea Junior Math Olympiad
Problems
(1)
AK_|_ BC wanted, 2 circles related
Source: 2020 Korea Junior Math Olympiad p2 KJMO
11/24/2021
Let
A
B
C
ABC
A
BC
be an acute triangle with circumcircle
Ω
\Omega
Ω
and
A
B
‾
<
A
C
‾
\overline{AB} < \overline{AC}
A
B
<
A
C
. The angle bisector of
A
A
A
meets
Ω
\Omega
Ω
again at
D
D
D
, and the line through
D
D
D
, perpendicular to
B
C
BC
BC
meets
Ω
\Omega
Ω
again at
E
E
E
. The circle centered at
A
A
A
, passing through
E
E
E
meets the line
D
E
DE
D
E
again at
F
F
F
. Let
K
K
K
be the circumcircle of triangle
A
D
F
ADF
A
D
F
. Prove that
A
K
AK
A
K
is perpendicular to
B
C
BC
BC
.
geometry
perpendicular