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KoMaL A Problems
KoMaL A Problems 2021/2022
A. 805
A. 805
Part of
KoMaL A Problems 2021/2022
Problems
(1)
KöMaL Geometry
Source: KöMaL A. 805
3/23/2022
In acute triangle
A
B
C
,
ABC,
A
BC
,
the feet of the altitudes are
A
1
,
B
1
,
A_1,B_1,
A
1
,
B
1
,
and
C
1
C_1
C
1
(with the usual notations on sides
B
C
,
C
A
,
BC,CA,
BC
,
C
A
,
and
A
B
AB
A
B
respectively). The circumcircles of triangles
A
B
1
C
1
AB_1C_1
A
B
1
C
1
and
B
C
1
A
1
BC_1A_1
B
C
1
A
1
intersect at the circumcircle of triangle
A
B
C
ABC
A
BC
ar points
P
≠
A
P\neq A
P
=
A
and
Q
≠
B
,
Q\neq B,
Q
=
B
,
respectively. Prove that lines
A
Q
,
B
P
AQ, BP
A
Q
,
BP
and the Euler line of triangle
A
B
C
ABC
A
BC
are either concurrent or parallel to each other.Proposed by Géza Kós, Budapest
geometry
komal
Euler Line