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Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2017 China Western Mathematical Olympiad
6
6
Part of
2017 China Western Mathematical Olympiad
Problems
(1)
Orthocenter condition
Source: CWMI 2017 Q6
8/14/2017
In acute triangle
A
B
C
ABC
A
BC
, let
D
D
D
and
E
E
E
be points on sides
A
B
AB
A
B
and
A
C
AC
A
C
respectively. Let segments
B
E
BE
BE
and
D
C
DC
D
C
meet at point
H
H
H
. Let
M
M
M
and
N
N
N
be the midpoints of segments
B
D
BD
B
D
and
C
E
CE
CE
respectively. Show that
H
H
H
is the orthocenter of triangle
A
M
N
AMN
A
MN
if and only if
B
,
C
,
E
,
D
B,C,E,D
B
,
C
,
E
,
D
are concyclic and
B
E
⊥
C
D
BE\perp CD
BE
⊥
C
D
.
geometry