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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2011 Korea Junior Math Olympiad
5
5
Part of
2011 Korea Junior Math Olympiad
Problems
(1)
ratio wanted related to orthocenter, circumcenter and 5 midpoints
Source: KJMO 2011 p5
5/3/2019
In triangle
A
B
C
ABC
A
BC
, (
A
B
≠
A
C
AB \ne AC
A
B
=
A
C
), let the orthocenter be
H
H
H
, circumcenter be
O
O
O
, and the midpoint of
B
C
BC
BC
be
M
M
M
. Let
H
M
∩
A
O
=
D
HM \cap AO = D
H
M
∩
A
O
=
D
. Let
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
be the midpoints of
A
B
,
C
D
,
A
C
,
B
D
AB,CD,AC,BD
A
B
,
C
D
,
A
C
,
B
D
. Let
X
=
P
Q
∩
R
S
X = PQ\cap RS
X
=
PQ
∩
RS
. Find
A
H
/
O
X
AH/OX
A
H
/
OX
.
ratio
geometry
circumcircle
midpoints
midpoint