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Problems
Contests
International Contests
APMO
1991 APMO
1
1
Part of
1991 APMO
Problems
(1)
Similar triangles involve centroid
Source: APMO 1991
3/11/2006
Let
G
G
G
be the centroid of a triangle
A
B
C
ABC
A
BC
, and
M
M
M
be the midpoint of
B
C
BC
BC
. Let
X
X
X
be on
A
B
AB
A
B
and
Y
Y
Y
on
A
C
AC
A
C
such that the points
X
X
X
,
Y
Y
Y
, and
G
G
G
are collinear and
X
Y
XY
X
Y
and
B
C
BC
BC
are parallel. Suppose that
X
C
XC
XC
and
G
B
GB
GB
intersect at
Q
Q
Q
and
Y
B
YB
Y
B
and
G
C
GC
GC
intersect at
P
P
P
. Show that triangle
M
P
Q
MPQ
MPQ
is similar to triangle
A
B
C
ABC
A
BC
.
trigonometry
geometry