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2018 IMO Shortlist
G7
G7
Part of
2018 IMO Shortlist
Problems
(1)
Just messy figure geometry
Source: ISL 2018 G7
7/17/2019
Let
O
O
O
be the circumcentre, and
Ω
\Omega
Ω
be the circumcircle of an acute-angled triangle
A
B
C
ABC
A
BC
. Let
P
P
P
be an arbitrary point on
Ω
\Omega
Ω
, distinct from
A
A
A
,
B
B
B
,
C
C
C
, and their antipodes in
Ω
\Omega
Ω
. Denote the circumcentres of the triangles
A
O
P
AOP
A
OP
,
B
O
P
BOP
BOP
, and
C
O
P
COP
COP
by
O
A
O_A
O
A
,
O
B
O_B
O
B
, and
O
C
O_C
O
C
, respectively. The lines
ℓ
A
\ell_A
ℓ
A
,
ℓ
B
\ell_B
ℓ
B
,
ℓ
C
\ell_C
ℓ
C
perpendicular to
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
pass through
O
A
O_A
O
A
,
O
B
O_B
O
B
, and
O
C
O_C
O
C
, respectively. Prove that the circumcircle of triangle formed by
ℓ
A
\ell_A
ℓ
A
,
ℓ
B
\ell_B
ℓ
B
, and
ℓ
C
\ell_C
ℓ
C
is tangent to the line
O
P
OP
OP
.
IMO Shortlist
geometry