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BxMO 2015, Problem 2
Source: Benelux Mathematical Olympiad 2015, Problem 2
5/10/2015
Let
A
B
C
ABC
A
BC
be an acute triangle with circumcentre
O
O
O
. Let
Γ
B
\mathit{\Gamma}_B
Γ
B
be the circle through
A
A
A
and
B
B
B
that is tangent to
A
C
AC
A
C
, and let
Γ
C
\mathit{\Gamma}_C
Γ
C
be the circle through
A
A
A
and
C
C
C
that is tangent to
A
B
AB
A
B
. An arbitrary line through
A
A
A
intersects
Γ
B
\mathit{\Gamma}_B
Γ
B
again in
X
X
X
and
Γ
C
\mathit{\Gamma}_C
Γ
C
again in
Y
Y
Y
. Prove that
∣
O
X
∣
=
∣
O
Y
∣
|OX|=|OY|
∣
OX
∣
=
∣
O
Y
∣
.
geometry
circumcircle