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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2005 Korea Junior Math Olympiad
5
5
Part of
2005 Korea Junior Math Olympiad
Problems
(1)
Perpendicular Line Geometry
Source: KJMO 2005
5/20/2018
In
△
A
B
C
\triangle ABC
△
A
BC
, let the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
hit the circumcircle at
M
M
M
. Let
P
P
P
be the intersection of
C
M
CM
CM
and
A
B
AB
A
B
. Denote by
(
V
,
W
X
,
Y
Z
)
(V,WX,YZ)
(
V
,
W
X
,
Y
Z
)
the intersection of the line passing
V
V
V
perpendicular to
W
X
WX
W
X
with the line
Y
Z
YZ
Y
Z
. Prove that the points
(
P
,
A
M
,
A
C
)
,
(
P
,
A
C
,
A
M
)
,
(
P
,
B
C
,
M
B
)
(P,AM,AC), (P,AC,AM), (P,BC,MB)
(
P
,
A
M
,
A
C
)
,
(
P
,
A
C
,
A
M
)
,
(
P
,
BC
,
MB
)
are collinear.In isosceles triangle
A
P
X
APX
A
PX
with
A
P
=
A
X
AP=AX
A
P
=
A
X
, select a point
M
M
M
on the altitude.
P
M
PM
PM
intersects
A
X
AX
A
X
at
C
C
C
. The circumcircle of
A
C
M
ACM
A
CM
intersects
A
P
AP
A
P
at
B
B
B
. A line passing through
P
P
P
perpendicular to
B
C
BC
BC
intersects
M
B
MB
MB
at
Z
Z
Z
. Show that
X
Z
XZ
XZ
is perpendicular to
A
P
AP
A
P
.
geometry
circumcircle