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KoMaL A Problems
KoMaL A Problems 2023/2024
A. 857
A. 857
Part of
KoMaL A Problems 2023/2024
Problems
(1)
Configuration with midpoints and altitudes
Source: KoMaL A857
10/11/2023
Let
A
B
C
ABC
A
BC
be a given acute triangle, in which
B
C
BC
BC
is the longest side. Let
H
H
H
be the orthocenter of the triangle, and let
D
D
D
and
E
E
E
be the feet of the altitudes from
B
B
B
and
C
C
C
, respectively. Let
F
F
F
and
G
G
G
be the midpoints of sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively.
X
X
X
is the point of intersection of lines
D
F
DF
D
F
and
E
G
EG
EG
. Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the circumcenters of triangles
E
F
X
EFX
EFX
and
D
G
X
DGX
D
GX
, respectively. Finally,
M
M
M
is the midpoint of line segment
O
1
O
2
O_1O_2
O
1
O
2
. Prove that points
X
,
H
X, H
X
,
H
and
M
M
M
are collinear.
geometry
komal