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Korea Junior Mathematics Olympiad
2020 Korea Junior Math Olympiad
4
4
Part of
2020 Korea Junior Math Olympiad
Problems
(1)
collinear wanted, altitudes, <QFD = < EPC, OH _|_ AQ
Source: 2020 Korea Junior Math Olympiad p4 KJMO
11/24/2021
In an acute triangle
A
B
C
ABC
A
BC
with
A
B
‾
>
A
C
‾
\overline{AB} > \overline{AC}
A
B
>
A
C
, let
D
,
E
,
F
D, E, F
D
,
E
,
F
be the feet of the altitudes from
A
,
B
,
C
A, B, C
A
,
B
,
C
, respectively. Let
P
P
P
be an intersection of lines
E
F
EF
EF
and
B
C
BC
BC
, and let
Q
Q
Q
be a point on the segment
B
D
BD
B
D
such that
∠
Q
F
D
=
∠
E
P
C
\angle QFD = \angle EPC
∠
QF
D
=
∠
EPC
. Let
O
,
H
O, H
O
,
H
denote the circumcenter and the orthocenter of triangle
A
B
C
ABC
A
BC
, respectively. Suppose that
O
H
OH
O
H
is perpendicular to
A
Q
AQ
A
Q
. Prove that
P
,
O
,
H
P, O, H
P
,
O
,
H
are collinear.
collinear
geometry