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Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2001 China Second Round Olympiad
1
1
Part of
2001 China Second Round Olympiad
Problems
(1)
An old geometry
Source: 2001 China Second Round Olympiad P1
8/11/2019
Let
O
,
H
O,H
O
,
H
be the circumcenter and orthocenter of
△
A
B
C
,
\triangle ABC,
△
A
BC
,
respectively. Line
A
H
AH
A
H
and
B
C
BC
BC
intersect at
D
,
D,
D
,
Line
B
H
BH
B
H
and
A
C
AC
A
C
intersect at
E
,
E,
E
,
Line
C
H
CH
C
H
and
A
B
AB
A
B
intersect at
F
,
F,
F
,
Line
A
B
AB
A
B
and
E
D
ED
E
D
intersect at
M
,
M,
M
,
A
C
AC
A
C
and
F
D
FD
F
D
intersect at
N
.
N.
N
.
Prove that
(
1
)
O
B
⊥
D
F
,
O
C
⊥
D
E
;
(1)OB\perp DF,OC\perp DE;
(
1
)
OB
⊥
D
F
,
OC
⊥
D
E
;
(
2
)
O
H
⊥
M
N
.
(2)OH\perp MN.
(
2
)
O
H
⊥
MN
.
geometry