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2023 ISL
G7
G7
Part of
2023 ISL
Problems
(1)
APMO 2010 P4 but you do it on all 3 sides
Source: IMO Shortlist 2023 G7
7/17/2024
Let
A
B
C
ABC
A
BC
be an acute, scalene triangle with orthocentre
H
H
H
. Let
ℓ
a
\ell_a
ℓ
a
be the line through the reflection of
B
B
B
with respect to
C
H
CH
C
H
and the reflection of
C
C
C
with respect to
B
H
BH
B
H
. Lines
ℓ
b
\ell_b
ℓ
b
and
ℓ
c
\ell_c
ℓ
c
are defined similarly. Suppose lines
ℓ
a
\ell_a
ℓ
a
,
ℓ
b
\ell_b
ℓ
b
, and
ℓ
c
\ell_c
ℓ
c
determine a triangle
T
\mathcal T
T
.Prove that the orthocentre of
T
\mathcal T
T
, the circumcentre of
T
\mathcal T
T
, and
H
H
H
are collinear.Fedir Yudin, Ukraine
IMO Shortlist
geometry