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KoMaL A Problems
KoMaL A Problems 2023/2024
A. 864
A. 864
Part of
KoMaL A Problems 2023/2024
Problems
(1)
Smoothie of incircle configurations from KoMaL
Source: KoMaL A. 864
12/12/2023
Let
A
B
C
ABC
A
BC
be a triangle and
O
O
O
be its circumcenter. Let
D
D
D
,
E
E
E
and
F
F
F
be the respective tangent points of the incircle of
△
A
B
C
\triangle ABC
△
A
BC
, and sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
. Let
M
M
M
and
N
N
N
be the respective midpoints of sides
A
B
AB
A
B
and
A
C
AC
A
C
. Let
M
′
M'
M
′
and
N
′
N'
N
′
be the respective reflections of points
M
M
M
and
N
N
N
across lines
D
E
DE
D
E
and
D
F
DF
D
F
. Let lines
C
M
′
CM'
C
M
′
and
B
N
′
BN'
B
N
′
intersect lines
D
E
DE
D
E
and
D
F
DF
D
F
at points
H
H
H
and
J
J
J
, respectively. Prove that the points
H
H
H
,
J
J
J
and
O
O
O
are collinear.Proposed by Luu Dong, Vietnam
geometry
komal