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Indonesia MO Shortlist
2015 Indonesia MO Shortlist
G7
G7
Part of
2015 Indonesia MO Shortlist
Problems
(1)
Geometry Collinearity
Source: 2015 Indonesia Math Olympiad Day 1 Problem 3
6/30/2017
Given an acute triangle
A
B
C
ABC
A
BC
.
Γ
B
\Gamma _{B}
Γ
B
is a circle that passes through
A
B
AB
A
B
, tangent to
A
C
AC
A
C
at
A
A
A
and centered at
O
B
O_{B}
O
B
. Define
Γ
C
\Gamma_C
Γ
C
and
O
C
O_C
O
C
the same way. Let the altitudes of
△
A
B
C
\triangle ABC
△
A
BC
from
B
B
B
and
C
C
C
meets the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
X
X
X
and
Y
Y
Y
, respectively. Prove that
A
A
A
, the midpoint of
X
Y
XY
X
Y
and the midpoint of
O
B
O
C
O_{B}O_{C}
O
B
O
C
is collinear.
geometry
circumcircle