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KoMaL A Problems
KoMaL A Problems 2021/2022
A. 828
A. 828
Part of
KoMaL A Problems 2021/2022
Problems
(1)
Orthocenter of triangle formed by polars is Nagel point
Source: KoMaL A. 828
6/11/2022
Triangle
A
B
C
ABC
A
BC
has incenter
I
I
I
and excircles
Ω
A
\Omega_A
Ω
A
,
Ω
B
\Omega_B
Ω
B
, and
Ω
C
\Omega_C
Ω
C
. Let
ℓ
A
\ell_A
ℓ
A
be the line through the feet of the tangents from
I
I
I
to
Ω
A
\Omega_A
Ω
A
, and define lines
ℓ
B
\ell_B
ℓ
B
and
ℓ
C
\ell_C
ℓ
C
similarly. Prove that the orthocenter of the triangle formed by lines
ℓ
A
\ell_A
ℓ
A
,
ℓ
B
\ell_B
ℓ
B
, and
ℓ
C
\ell_C
ℓ
C
coincides with the Nagel point of triangle
A
B
C
ABC
A
BC
.(The Nagel point of triangle
A
B
C
ABC
A
BC
is the intersection of segments
A
T
A
AT_A
A
T
A
,
B
T
B
BT_B
B
T
B
, and
C
T
C
CT_C
C
T
C
, where
T
A
T_A
T
A
is the tangency point of
Ω
A
\Omega_A
Ω
A
with side
B
C
BC
BC
, and points
T
B
T_B
T
B
and
T
C
T_C
T
C
are defined similarly.)Proposed by Nikolai Beluhov, Bulgaria
geometry