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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1997 Mexico National Olympiad
2
2
Part of
1997 Mexico National Olympiad
Problems
(1)
4 points defined be equal ratios in a triangle, collinearity wanted
Source: Mexican Mathematical Olympiad 1997 OMM P2
7/28/2018
In a triangle
A
B
C
,
P
ABC, P
A
BC
,
P
and
P
′
P'
P
′
are points on side
B
C
,
Q
BC, Q
BC
,
Q
on side
C
A
CA
C
A
, and
R
R
R
on side
A
B
AB
A
B
, such that
A
R
R
B
=
B
P
P
C
=
C
Q
Q
A
=
C
P
′
P
′
B
\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP'}{P'B}
RB
A
R
=
PC
BP
=
Q
A
CQ
=
P
′
B
C
P
′
. Let
G
G
G
be the centroid of triangle
A
B
C
ABC
A
BC
and
K
K
K
be the intersection point of
A
P
′
AP'
A
P
′
and
R
Q
RQ
RQ
. Prove that points
P
,
G
,
K
P,G,K
P
,
G
,
K
are collinear.
ratio
geometry
collinear