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Rioplatense Mathematical Olympiad, Level 3
2001 Rioplatense Mathematical Olympiad, Level 3
5
5
Part of
2001 Rioplatense Mathematical Olympiad, Level 3
Problems
(1)
Sum is 180
Source: Rioplatense Olympiad 2001
2/28/2018
Let
A
B
C
ABC
A
BC
be a acute-angled triangle with centroid
G
G
G
, the angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
intersects
A
C
AC
A
C
in
D
D
D
. Let
P
P
P
and
Q
Q
Q
be points in
B
D
BD
B
D
where
∠
P
B
A
=
∠
P
A
B
\angle PBA = \angle PAB
∠
PB
A
=
∠
P
A
B
and
∠
Q
B
C
=
∠
Q
C
B
\angle QBC = \angle QCB
∠
QBC
=
∠
QCB
. Let
M
M
M
be the midpoint of
Q
P
QP
QP
, let
N
N
N
be a point in the line
G
M
GM
GM
such that
G
N
=
2
G
M
GN = 2GM
GN
=
2
GM
(where
G
G
G
is the segment
M
N
MN
MN
), prove that:
∠
A
N
C
+
∠
A
B
C
=
180
\angle ANC + \angle ABC = 180
∠
A
NC
+
∠
A
BC
=
180
geometry
angle bisector